I am not sure why you call this phenomenon a dissonance. The fact that the same idea often pops up in different places in mathematics shows, in my view, how harmonious the mathematical edifice is. I may recommend a very nice book How The Other Half Thinks by Sherman Stein. Stein explores verious circumstances that could be described by binary strings. Of course, group or matrix theory, or differential equations provide numerous addtional examples.
At this site, there's a recent page where the same system of equations describes two quite unrelated problems:
You may also find useful D. Wells' You Are a Mathematician.
The matter of the multitude of infinities is somewhat different. However, Cantor's diagonal process has been used by Güdel, Turing, Chaitin and probably by others to prove different impossibility theorems. Chaitin's The Limits of Mathematics provides a lively reading.
You may want to look for examples of real mathematics that requires very little mathematical knowledge. (This is what Stein's book is about.) At this site, you may want to check, for example,
But there is more.