Question: My probation period is ending soon, and my boss asked if I want to continue with the technology I was using, or to change to other one.

I have to announce my decision on the last day of the probation period. Changing doesn't make a lot of sense, which makes me wonder if I choose wrong, will that be my last day.

So my question is is it common and widely accepted to test person's worth and motivation by setting this kind of tests?

(This is not a company-specific question, but more like "do these kinds of habits and company cultures exist?").

Answer: 
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> This is not a company-specific question, but more like "do this kind of habits and company cultures exist"
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Clearly such things do exist or you wouldn't be asking the question but in my 30+ years in the workplace, I've never heard of it. It's an interesting question to be sure.

I can understand why you would be suspicious of it as a trap, especially if you have to "announce" a decision rather than a preference.

Were it me, I'd answer honestly. It may well be a trap but there's not a lot you can do about it. Answer with your honest choice but follow it up by saying, "but I understand that the needs of the company obviously are important and so I'm willing and happy to go where needed."

Make it clear that you're a team player and you'll go where they need you to go if you're willing to do so. 

Something else you might consider is the place of the "other one" in the company. Your boss could be offering you an opportunity that you might otherwise have. For instance, if you're working on a technology that is still well-used but the company has plans for something different, that might be something to consider. For instance, if you're an Microsoft person but the company will be transitioning to LAMP, they could both have a strong need for the future, but the longevity of the other could be greater.

Also, perhaps your boss is testing you in a way you hadn't considered, giving you the opportunity to ask "where am I most needed?" or maybe even a simple, "what do you think?" to show that you value (or at least consider) his opinion. I could definitely see a question like that being used to see how focused you are on company needs versus personal desires.
Having never met your boss, it is hard to gauge his motivations. But let's look at this logically. Your employer hired you for a reason. They need the work done. The probationary period is only there to give them a way out in case they made a mistake in hiring. 

If you've been a good employee and lived up to their expectations of you, it would be extremely foolish of them to kick you out just because you expressed an interest in continuing work in a technology you know. Good talent is surprisingly hard to find. Replacing you would be difficult and expensive for them. This would not be a good move on their part.

If, on the other hand, your work has not been up to snuff and they are planning to let you go, they wouldn't be giving you a test to try and save yourself. The probationary period you've gone through is there expressly to give them a way to rid of you if they want to. They don't need to come up with an excuse to do it by testing you, they're just going to kick you out.

So I think it is very unlikely this is some kind of trap. It is more likely that your boss has already decided that you have passed your probation and wants to keep you. He is now looking to your future giving you a choice as to whether you want to keep doing what you are doing or try something new. It could be he waited until the end of your probation because he knows that learning a new technology is hard, you are likely to stumble, and didn't want that to be a reason for you to fail.

The thing to do is have an honest discussion with your manager. Express your concerns and always remember that you have control of your career. But hear him out too, because he probably has a good reason for asking you to change. Then you can make your decision.

As a side note: It sounds like you are working in technology, so I will caution you about getting too attached to any technology you are working with now. Our industry is extremely fluid and anything you are doing now is going to look completely different in 5 years (or less). By all means, take the time to master your skills, but in the long run adaptability is far more valuable.