What should I do while in an undergraduate engineering course to keep a proof of the math knowledge I gain by self-study?

I feel sufficiently compelled to answer this as a recent math and physics major who completed his degree requirement.

In my undergraduate class, I was taking various combination of 4 math and physics units each semester -- 4 unit is the heaviest load one can take in a semester. At the top end they were partial differential equations, real analysis, Abstract Algebra, Topology and geometry, mathematical physics, Advanced topics in Quantum mechanics and atomic physics such as quantum mechanics operators, measurement theory, spin and orbital momentum, LS coupling, jj coupling, helium atom, fine structure, hyperfine interaction, atoms in magnetic fields, electron spin resonance, transition probabilities, astrophysics and condensed matter physics, dynamical systems, applied and computational modelling of physical systems. On top of these, I took various quantitative units in economics and did very well even in absence of the prerequisites.

, denotes a limiters of the different subjects/ units/ course/ modules

This is a 3 years standard bachelor but I completed it in 4 years. If you are prepared for the sheer amount of hard work, extreme burn out and a lower than average GPA, do it. Mine dipped slightly below a 3.0/4.0 but I had strong references and computational skills.

The natural thing to do in such a case would be pursue a minor in mathematics. But, unfortunately, our university does not offer any minor degrees or dual major degrees. So, it's not possible for me to formally take extra classes in mathematics.

Everything on your transcript is just ceremonial. What is more important is you knowing the subjects and actually being able to demonstrate it on a technical test/ chalkboard when called.

I'm speaking this as someone who has recently started collaborate on a research project with a professor while looking around for industrial opportunities.

If you're looking to work alongside research members, academics or professors, you may be asked to provide some insight as to what you have already covered in your time; in this case, you are free to draw upon what you have taken in your undergraduate curriculum and free time. People effectively wants to know what you know, not what you have taken.

In my limited experiences, professors are actually impressed with students who challenge themselves by taking advanced units, even if they consider you foolhardy.

Upon pondering a bit I realize that I might want to pursue my higher studies in some interdisciplinary area which involves knowing things from electronics engineering as well as from the rigorous mathematical physics and statistical learning (machine learning/data science/AI). I'm not sure if such an interdisciplinary area of study even exists at the graduate level (?).

While I am equally unfamiliar with such combination in academia, you may circumvent the lack of such opportunities in universities through independent study. Most research these days are cross functional across seemingly unrelated fields so much is dependent on the candidates to independently learn. In my short experiences with looking for opportunities in areas of data analysis and machine learning, the common litmus test is a request for candidates to undergo a technical test much like how developers are subject to technical tests despite their year of experiences.

In STEM, hardly anyone will give you the luxury of knowing something just because something came up on your transcript.

But I'm really enjoying learning the new things in mathematics and I don't want my spending time on learning these things go in vain.

If this is not a contradiction then there is insufficient data.