The game you describe is a
sibling close relative of Schuh's Divisor Game, which is played on the set of all divisors of a positive integer $N$. For a description of the game, see Problem 12 of
This is the first problem set in an advanced undergrad number theory course I taught twice at UGA in recent years. If memory serves, I got the problem from Aaron Abrams, who used it when he taught the same course at UGA a few years before. It is absolutely one of my favorite problems in the whole course: I don't want to say that if you don't like this problem you're not going to like number theory or mathematics (but part of me feels this way!). Note that a paraphrase of part d) is that the unjustly more famous combinatorial game Chomp can be simulated by the divisor game. Since Chomp is already unsolved, so is the divisor game. Let me also remark that I actually had a student who wrote a computer program implementing and playing the divisor game for a final project in the course. She did a very nice job!
Schuh's divisor game can also simulate the subset take away game: let $N$ be a product of $n$ distinct primes.
[Added: I just looked more carefully at the rules of your game. I formulate these games in the misère version, meaning the person to make the last move loses. However it is equivalent to disallow the ability to take away everything at once, i.e., in this case to take the divisor $N$ of $N$ and all of its divisordivisors. In the coming generalization, this means removing the maximal element from the poset. If the poset also has a minimal element, as it does for me, then a classic strategy stealing argument shows that the first player must have a winning strategy: he could just become the second player by taking the minimal element. However, in your version the minimal element is removed from the start, so it is equivalent to the first player having already played and taken the least element. It is easy to see that in the for general subset game $N$ this need not be a winning strategy, e.g. consider the case of $N$ a prime power. For the case of squarefree $N$, as you say, it looks more likely that this is the beginning of a winning strategy.]
Still, it is interesting and useful to think about a yet more general game, the poset game which is described as problem (G2) in this set. (Note that one could get away with weaker finite finiteness conditions than just the finiteness of the entire poset, but I didn't want to push that point in a first number theory problem set.) The game that you describe is more visibly equivalent to the poset game played on the power set of a finite set, partially ordered by inclusion (or, indeed, by reverse inclusion).
You will see there that there has been another interesting science project math paper written about this subject, namely the following one by Steven Byrnes:
There is a nice theorem here, the poset game periodicity theorem!
In general, I find this to be an attractive corner of mathematicsand am surprised that more people don't know about it. It certainly seems like there is plenty of further work to be done.