Is there a place in academia for a physicist who reads mostly about math?
I have a somewhat similar background: I primarily studied physics but also did roughly a bachelor in mathematics (a “Vordiplom” to be precise). I never liked regular theoretical physics much because the mathematical background was never properly explained, if it was mentioned at all. I did my master’s-thesis equivalent and PhD in complex systems / chaos theory.
So far, my mathematical background was an enormous benefit to me. I published three papers which clearly wouldn’t have been possible without it, and in the rest of my work, it often helped me to understand things much quicker. The main impact of mathematics on my work is that it allows me to identify and formulate problems, draw connections, know the keywords to find the relevant literature and understand it. While I did prove things at times, this was only the icing on the cake – and I would consider most of my proofs not very challenging. The main exception was the proof of a number-theoretical statement that I had already confirmed very well empirically. If I had not eventually found the proof, I would probably have consulted the next-best expert on number theory at my university. Still, I am more proud of finding the right statement to prove than of proving it. Almost everything that I proved is published.
So, I really do not see a problem with your interests. There is almost certainly a workgroup that is interested in the same questions, though it does not seem to be the one you are in right now. However, if your supervisor accepts your mathematical inclination, they may also be able to connect you to the people who care about these issues to give you feedback.
Bringing a new background to your workgroup is an asset that is very likely to make you see new connections or identify and solve new problems, which can eventually lead to new physics. You may need a proof on the way, but I would not worry that this will consume most of your time. At the end of the day, even most mathematicians do not spend all day at their desks trying to prove something, but also have to understand existing concepts, find connections, and so on. And should you stumble upon something that you cannot prove, you can still collaborate with mathematicians or leave it as an open problem/conjecture.
Another perspective could be to simply establish a mathematical framework for what your workgroup does (which then may contain open questions for you or others to fill). Depending on the relevance and extent of what we are talking about, this may already be worthy of a PhD or even enough work for an entire research career.
Finally, some random asides:
You probably are at a stage of your career and a field, where few people care about whether you are actually a mathematician or a physicist.
Beware of the impostor syndrome. If your master’s thesis could be turned into a paper, that’s already a good start. That somebody else managed to publish it earlier, is bad luck for you, but still shows that you produced publishable work.
If you are doing a lot of programming work, it can be very helpful to understand more about the methods you are using. Moreover, you may find yourself developing new numerical methods – where a mathematical background is extremely helpful.
a physicist should know enough math to be able to do his research, and no more
While there is some truth to this for some areas of physics, most physicists do not actually know enough math to do their research, or they are limited in what they can research by the math they know. Most importantly, in the more mathematically inclined fields of physics, I disagree with this due to the reasons stated above.
So far, you've been reacting. You sign up for a course, maybe it's required, maybe it's not. You're given a homework assignment. The hand waving approach in the lectures bothers you and you spend a lot of time building up the mathematical background, and you fill in the blanks. If you successfully managed to keep your instructors happy enough to give you good grades while you were engaged in this intensive self-study, good for you!
At some point in the not too distant future, you are going to want to start taking the initiative to give shape to your studies. Here are some ways to do that:
Search for papers that really, really intrigue you. One of these might give you an idea for a direction you'd like to go; at the very least, such a paper can give you a mental image of what a math-friendly physics project can look like.
Find out how you feel about other aspects of physics. You mentioned that you enjoy working out the mathematical underpinnings of the assignments you're given, and you enjoy programming. But I wonder if you have figured out how you feel about working in a lab, designing experiments, carrying them out, analyzing experimental results, writing programs to analyze experimental results, choosing and installing new equipment, designing a new or modified experimental apparatus, communicating with the engineer and the machine shop technician, testing a new experimental set-up ("commissioning").... This is good information to have about yourself.
Please don't forget that you are studying in the Third World, not that one should look down on science in the Third World, but just to reassure you that it is forgivable that your instructors might do more hand waving and less rigor than is at least sometimes found more commonly in the developed world.
My spouse is an experimental physicist, who uses math very, very frequently, and sometimes writes a theoretical paper, using mathematics to explain, as you have been doing so far, but also to create, as you are dreaming of doing.
Note 1: At some point you may want to correspond with an author of a paper you've identified in your search. Science as become incredibly international.
Note 2: I wasn't sure from your post, whether you might sometimes have a bit of trouble keeping your eye on the big picture (getting the assignments done, without getting so sucked into the mathematical underpinnings that you might have trouble making your deadlines). If so, take a look at Is there a place in academia for someone who compulsively solves every problem on their own?, which is a rather extreme example of what I was describing. If not -- have fun in your explorations!