Most equivalence relations eventually drop any special notation in favor of =

In general, = is not rigorous. It's used as a stand-in for the word "is".

Another example is when someone writes x = 1, ..., N, meant to imply x is changing, not that x is equal to a tuple. Also the index in summation notation, which really is an assignment as in programming. I could go on.

How about Σ Sigma notation for summation? It's often written with something like i=1 at the bottom and 1000 at the top to denote the summation from 1 to 1000. The Wikipedia article at https://en.wikipedia.org/wiki/Summation has some examples.

Defining things, usually preceded by "let", "if", etc.: "let f(x) = 2x+1" or "let w = 2", etc.

Sometimes mathematicians us ":=" to be clearer and shorter.