Did the professor "give" me this project?

It's research. You set out to build a plane and end up with a space ship. Or a glider -- most often a glider. If you were my student, I wouldn't say "I gave you the problem". I would probably say that I started you on a research direction.

Two years ago, a friend of mine gave me an idea and I tried to follow it. His idea seemed to lead nowhere. At some point, I read some paper I found by accident and found a way to continue his idea that he most likely couldn't have thought about. I did calculations and wrote a paper about it. Because he gave me the starting idea and constantly provided feedback and suggestions, I put him as last author on that paper.

Usually, advisers get authorship that way. In other words, they start you on a problem, and as you progress, they try to guide you and suggest you resources that could help you. You do start with their idea, which is often wrong, you do the work, and get something that is new. If it comes to authorship, some think that it's enough to get it if they just proposed the problem to you and did nothing else. I think that's not ethical, but it's hard for a student to enforce ethics.

The situation is different if the adviser has some strong indication of what the result of the proposed problem should be. That means he already did some work on the problem and needs you to finish. Then he deserves authorship even if he does nothing but point the problem out to you.

So, "giving the problem" and giving you a direction to work in is quite different. In the first case it's like you're a miner, and your boss tells you to continue digging a given hole because there's gold in it. You find the gold, he deserves a cut. But suppose he tells you to dig a random hole. You find no gold, you move to another hole, and find it there. He doesn't deserve a cut in that case, but he deserves thanks for letting you know that digging in a hole might lead to gold.

If it was business, the professor would get nothing beyond thanks, because they usually say that ideas mean nothing, it's only the implementation that counts.

A short answer: first, in mathematics, this is a typical way of speaking, yes. At best, the advisor has a very good idea that the suggested direction will be fruitful, in one way or another, and is not asking for co-authorship (despite traditions in other disciplines).

That is, hopefully it's much more than "just an idea", namely, it is an idea that fits into or extends something that the advisor has thought about for a long-enough time so that the way it should play out is approximately predictable. This is very important side-information! :)

So, yes, I myself "give" my students projects... based on a large context... which I would bet will turn out well, one way or another... and be fairly interesting to experts, and be publishable, and not tend to allow them to be "scooped", and be complete-able within a good time-frame. All those attributes are highly non-trivial (as we say in math) to arrange. Some expertise required.

As a mathematician, I personally wouldn't care whether I "gave" a problem to anybody or whether anybody "gave" it to me. This means pretty much nothing. The mathematical problems are not owned or exchanged for profit. They are just shared in the hope that somebody may turn out smart enough to solve them. So if your adviser wants to be able to say that he gave the problem to you and you want to declare that you picked it yourself, you two may have as endless argument about it as you wish, but many people would just shrug at all that nonsense.

As to bringing the matter further to the matters of co-authorship, etc., in mathematics, if the professor is any good, he wouldn't insist on, or even suggest a co-authorship to his student unless he did the lion's share of the work himself. He may agree to it if the student offers it though. The standard politeness requires to be generous with offering co-authorships and reluctant with accepting them.

If all your adviser really wants is that you thank him in the paper for attracting your attention to that problem, I would certainly oblige. This costs you nothing and doesn't diminish your credit by any amount (if you care about such things) while it can help him a bit with his promotion score or grant proposal.