In fact, every number base is base "10" when you interpret the "10" in that base.
10 binary is two.
10 octal is eight.
10 hexadecimal is sixteen.
This is the very definition of a number base: it is the multiplier that you represent by appending "0" to a string of numeric characters in that base.
So that is where the fundamental and special nature of "10" comes from; it's not because it happens to mean "ten" in our customary number base.
Ten is nothing special, "10" is. "10" is simply the way you write N where N is whatever number base you're working in. It's just as special in every number base!
p.s. I'm sorry you were downvoted so heavily for asking an honest question. You can't be the only one who has wondered about this, and your question led to an interesting discussion.
Your reasoning seems quite obvious now with hindsight after having given it some thought; and yet, even still I have such a strong inclination that ".........." is a special number? Why? I guess it really is entirely my cognitive bias, because I can't find a reason for it.
But I should have known better as I've--probably obviously being a user here--encounteted binary more than just a few times. And even still, it never occurred to me that 10 in binary is just ".." And then 100 in binary is 1 order of magnitude of "..," 4. And 1,000 is just two orders of magntiude, 8. But still, intuitively this does not seem as natural as 10, and I guess that is completely cognitive bias.
Am I retarded to not have realized this? Maybe, but I actually was so curious about this that I tried to quiz some colleagues by asking what 1000 and 1001 is in binary and only one person got it right immediately, probably by understanding orders of magnitude and not by rote memorization. All the others got it by counting in binary, and one final person was annoyed and questioned why I was asking about binary (oops, sometimes being inquisitive is not socially acceptable). By the way, I work with app developers, most of whom do not have backgrounds in computer science, same as myself.