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Lots of issues to look into this article by Foreign Affairs, the first being that the wording around the article itself.
To elaborate, using the term “soft power” implies weaponization of a subject, typically used when discussing a nations influence. To attribute this to a religion, namely Islam, makes sense as the religion has been vilified in the media over the last 2 decades in order to manufacture consent for the united states wars in the region. 
Now, not only does using “soft power” imply the religion is some weapon, but it also does this really common thing I notice in western media: painting the religion and it’s followers as a monolith.
I can see other comments around this topic and I honestly expected it considering reddit’s main demographic.
tldr: the article is nonsensical clickbait material using the low hanging fruit of islam hate bait to get clicks, this feeds into redditors confirmation bias
I've never heard of this phrase anywhere else. I don't think Islamic culture is exalted anywhere outside of Muslim countries (much of the west finds it despicable), so I doubt they have much soft power to speak of. >

I would argue that islamic influence in general is giant negative on any society. islam is an extremely violent, authoritarian, oppressive, and extremist ideology. Many people, especially in the West, can't seem to comprehend just how far out there islam is, there's nothing in the West to compare it to besides itself. 
Now, the reason why islam is all of these things is because islam operates through fear. If you leave islam, you will be killed. If you criticize islam, you will be killed. If you get caught stealing, you'll have your hand cutoff. If you're a woman and you disobey your husband then you will be beaten. If you don't conform to islamic teachings then Allah will punish you by burning you in hell for eternity. Everything about islam functions through the lens of fear. 
islamic cultures are built around this ideology, and so they're going to be reflective of its values and teachings. Of course, not all islamic cultures are the same, they vary wildly from place to place. However, there's a key trend among all islamic cultures. The more in touch with islam a culture is, the more extreme, violent, and oppressive it is. For example countries like Afghanistan, Pakistan, and Saudi Arabia are more in touch with the actual teachings of the islamic scriptures (quran and sahih hadiths) and therefore, their governments and cultures are reflective of that. 
However, when islam isn't the dominant ideology, but is only a broad secondary ideology, things are different. For example, countries like Turkey, Albania, and Bosnia are all islamic, however, they are less in touch with the islamic scriptures and instead have their socities driven by more modern, more secular ideologies. Because of this they are less extreme, less oppressive, and more free and open. 
The more influence islam has on a society, the more it heads in the direction of countries like Afghanistan, like Saudi Arabia, like Algeria. A good example of this would be Indonesia. For the past decades, Saudi Arabia has been trying to do everything it can to get Indonesia to become less secular and more islamic, and they're succeeding. Indonesia over the decades has slowly but surely gotten less and less free. I mean just recently the islamic parties pressured the government to pass legislation that banned adulatory.
I see islam's influence as a negative because whenever it is put against a competing ideology, it is always the worst one. This is best seen when two very similar neighbors decide to take different routes. For example, Algeria and Tunisia. Sample people, same history, same language... yet very different worlds. Tunisia decided to go down a more liberal, more secular path, while Algeria decided to go down a more islamic path. The results are what you would expect. Algeria became a backwards, authoritarian country akin to Saudi Arabia (minus the oil money), while Tunisia became by far the freest country in the entirety of the Arab world (according to the freedom index). Tunisia even became the first country in the Arab world to elect a woman as PM. There are other examples like this like Algeria and Morocco, Pakistan and Bangladesh, Turkey and Iran, and so on.
The point is that the less influence islam has on a society, the better it does. islamic soft power is something that's negative because of what islam is, a tyrannical ideology. When a society allows a extremist, oppressive ideology that enforces itself through fear and violence take hold, then it's going to suffer and decline. I think countries should actively combat and reduce islamic influence, not embrace it. islam is something that holds societies back. >

I think what you’re getting at here is partly the difference between Islam and Islamism, the latter being a political ideology and movement. 
Being Turkish in Turkey is tied to professing Islam and speaking Turkish. Since the founding of the state Islamic faith was part of how the nation was defined even as the ideology of the state was espousal of laicite (Kemalist secularism is beyond what most people understand as secularism in English). In the case of Algeria, Christians and Jews were largely expelled as part of the process of creating a state. This was partly tied to decolonization and prior French divide and conquer strategies. Berbers in Algeria and Kurds in Turkey also present major Muslim minorities which complicated building a state. It’s the political path afterward that’s defining what you’re talking about. 
Turkey and Indonesia are both large Islamic societies (in fact both among the most populous). The influence of Islam on their society is essential.  Islamism is not necessarily an inevitable outcome of their embrace of that fact, nor does their existence outside of Islamism make them any less Islamic. 
While Turkey under Kemalism embraced secularism while Indonesian Pancasila with monotheism is a form of restricted pluralism. I’d say that roughly spans the religious policies seen around the rest of the world (apart from atheism) where one also seems religious and moralistic movements akin to Islamism. The main difference is that Islamism has in recent decades been ascendant across much of the Muslim world. >

Articles pay walled so didn't read it. That said use of religion as a way to influence foreign actors has been going on since like forever. So I'd be interested what the new phenomenon of 'Islamic Soft Power' is. >

AFAIK the most reasonable example given the subject would be using oil supply manipulation to affect the world economy and the "soft power" coming from having a degree of control over the resulting choices that must be made. What does culture have to do with it? >

Islam is reviled in the West, India, Russia, China and southern Africa. Outside of Islamic countries it has basically no influence. >

I think the comments so far here are looking at the Islamic world as something timeless or out of time when it has transformed a lot in the recent decades and centuries. This is a huge mistake. 
I will speak to a couple things I am more familiar with. I am not Muslim but I have lived in places where Muslims were a major minority and local majority. One thing that has struck me when speaking with Muslims born and raised in the West at least is how different the Muslim experience is inside and outside a majority-Muslim society. Most prominently is how much the call to prayer sets the rhythm for the day. I have known Muslims who make space strictly for the same pattern in an office setting with few Muslims but it is not the same as when society broadly is observing it. As a non-expert outsider I have the impression that Islam has well established tenets to work both ways, as a minority faith and as essentially a blueprint for a complete society, my impression is the latter is rapidly changing. 
I am two generations removed from the immediate experience of it but a major change in the Islamic world underway is the decreasing proportion of non-Muslims in it. Being from a non-Muslim community from the Muslim world is defining for all of these communities. To offer prominent examples: Copts, Jews, Mandeans, Armenians, Assyrians, various Arab Christian groups, Yazidis, these groups have been present alongside Islam since the time of the Prophet and are steadily leaving the Islamic world or being killed or converted sometimes to a point of essentially total disappearance. There’s similar processes in South Asia but dhimmi status there is less clearly established and it’s less familiar to me. The process has aspects of nationalism and colonialism tied up with it, but I think it’s likely extremely impactful on how Islamic society is defined since it’s often designed to make space for these groups.  
Inside the groups we understandably are concerned first and foremost with the preservation of our own communities, but I’m often disappointed that the impact of these changes on the Islamic world seems to be something of a taboo. The exception to this I’ve seen is Lebanese people where a mutual experience of hardship by all faiths allows space for conversation even if it’s fraught. It’s not really the Muslim world per se, but in the Balkans where conflict was also multifaceted and remains raw at best similar conversations also happen. These are in fact among the communities I’m most familiar with so perhaps there are similar conversations I’m just not seeing, but it would be unbelievably tragic if long wars were the only path to these conversations. 
I’d be curious to know what Muslims think. The Islamic world my grandparents generation lived in is radically different from that my parents travelled or any that I’ve seen in my life. I strongly suspect that the disappearance of non-Muslim minorities has played a role in that transformation in ways not always anticipated when it was happening.