What is the Senju clan's speciality? Is it wood style? If yes, then how come only Hashirama can use it?

There has to be some special power that all the members of the Senju clan possessed, or else how could they be the greatest rivals to the Uchiha (who had Sharingan) who defeated every other clan other than the Senju?
The only Senju to showcase the Wood Release was Hashirama Senju. You could say that the Wood Release was Hashirama's speciality. Even his brother Tobirama did not use Wood Release all that much, but instead had a variety of speciality techniques including Space-time jutsu, Edo Tensei, Kage Bunshin, etc. 

The Uchiha are adept at using fire techniques, because of their natural affinity towards it. The Senju are commonly called as "Senju clan of the forest". So we could only assume from the name that they are adept at using forest related skills; Wood Release. 

But from the Senju we have seen in battle, namely Tobirama, Hashirama and Tsunade, the common feature about them is their unique fighting style and skills. This would lead into a more agreeable assumption. The clan's name, "Senju", literally means "a thousand skills". So this would mean, rather than the whole clan specializing in one type of jutsu/technique/nature release/skill, the clan members would be specialized in different types of jutsu. This actually fits the meaning of their name. Which means, there isn't a trademark clan ability. This assumption is supported by the Senju we've seen in battle multiple times.

The Uchiha hadn't defeated all the other clans. It was just said that they were among the top powerful clans, only challenged by the Senju. The Uchiha and Senju had been fighting against each other over centuries, even before Hashirama and Madara's times. This was because of the Indra and Asura feud. So while the Uchiha were powerful, the only clan able to go against them were the descendants of Asura. That's the main reason as to why they could go toe to toe.

TL;DR : The Senju clan doesn't have a trademark ability. But all its members have proficiency in different types of jutsu. Because of being the descendants of the Sage of Six Paths' sons, the Senju and Uchiha were able to go toe to toe against each other.