As you start to become more experienced in your studies, you will see patterns. Mark Twain said, history doesn't repeat, but it rhymes. I think this is the right spirit to have, looking at analogies. Not precision (2:3 as 4:6). But general similarity.
Learning different things not only gives you a storehouse of examples, but also trains your brain at finding patterns (the act of looking for and finding analogies, partial analogies, FALSE analogies [I once thought Excel pivot tables were just like a third dimension...good guess, but no]). By looking for analogies, you are an active participant in the process.
I wouldn't belabor it. But just be mildly open to similarities. E.g. hydraulic resistance in a fluids circuit has mathematically same parallel and series addition to electrical resistance. (Like that one's always helped me because when an electrical circuit confuses me, I think of pipes...and visa versa.) Somehow having two physical models AND the math abstraction, helps me to internalize the concepts. Like being three blind men at an elephant instead of one.
Maybe a less direct analogy would be considering fluorine chemistry. Sometimes, you can make a prediction based on the analogous chloride (downstairs brother), or sometimes the analogous oxide (next door brother). Or sometimes different from both. But sure as shit, you got a hell of a lot more insight if you understand periodic trends than if chemical combinations are pure chance.
And I wouldn't totally underestimate the value of a storehouse of information either. You will see a lot of people dismiss that as "just memorization". But to be a Renaissance man, you do need to know a few things first! If you want to use the insights from catapult construction to help your painting and visa versa, you need to learn some of each!
Now, yes, it is also good to have puzzle solving experience. Good in general. And good even for the purpose of learning the basic materials (progressive struggle drives retention and comprehension...why the worse athletes are sometimes better coaches). But I wouldn't completely fetishize puzzle solving and deprecate information knowledge. Feynman was renowned for looking at problems originally. But he also ran through very significant amounts of very normal drill (and recommended it to struggling students at Cal Tech). You can look at his calculus notebooks from high school and see both original thinking AND basic drill (LOTS of it). Furthermore, before he did his most significant work, he realized he needed to buff up his knowledge of classical E&M (which is a ballbuster of a subject, often harder than some of the sexy topics). And he grabbed a normal text and worked every problem.
So, net, net. I think there are wonder puzzle solving AND knowledge learning opportunities in a conventional course of study. Please do not underestimate the huge value in learning classical mechanics (with the Lagrangian and all) before dealing with the "Internet-sexy" abstractions of quantum mechanics. If you do other stuff on the side, great. But only if you are crushing the normal course. If you get below a high A in the normal courses and run around on reddit and sites like this working on popular books on relativity and QED, you're making the wrong choice. Those basic courses have huge "meat" to fuel more creative later research work. Both the information in them and the training of your mind involved in mastering them.
