There is a sense in which it is not fair to say that one "removes structure" when one considers a more general class of mathematical objects. Usually one tries to do analogous things with the more general class that one was able to do with the more specific class, and this requires new definitions and techniques.
Let's take the example of "removing the geometry from Euclidean space". Just forgetting about distance and angles etc. is easy to do. But if I want to actually do anything in this new geometry-less world, that's another story. Let's say, as in the question, that I've moved into point-set topology. First need to know what a topology is, and that takes a certain level of mathematical maturity to grasp. Once I understand what a topology is, I will probably learn that there are many different possible topologies on a set. I will also learn that the notion of "distance" I used to have in Euclidean geometry gives one such topology. Then I can check my understanding by convincing myself that some of these new notions I've learned about really do specialize to the things I used to do when I was doing geometry.
This is a common pattern in mathematics: generalize with an eye towards recovering old results or analogs of old results in the generalized setting. This almost invariably leads to a net gain in number of possible structures, and these possible structures are the things that wind up being studied.
On a more concrete level, Jessica B's comment also addresses a difficulty that can occur when removing structure: we can't use the rules we used to use. Consider a linear algebra student encountering matrices and matrix algebra for the first time. The student would like to do algebra with matrices just like with numbers. But now there are things that are not invertible, and even when a matrix is invertible the student can't "divide" both sides of an equation by a matrix because they don't have commutativity. So the student needs to adjust the way that they've always done algebra.