There are two types of programming language. The most common type (including C) could be described as "imperative", i.e. most of the source code contains commands telling the computer to do something. The "=" sign is just as shorthand for "work out the value of what is on the right hand side, and then assign it to the variable on the left hand side". As you said, this isn't what "=" means in mathematics, but there is a limited set of characters available on a standard computer keyboard.
There are a few languages that are "declarative", where the source code is more like a "set of equations" that the computer "solves" to produce a result (or several results, if the solution isn't unique). These languages are usually less general-purpose, because the method for "solving the equations" has to be built into the implementation, so you can't just throw anything at it and hope it will figure out how to solve it. PROLOG is an example of that type of language.
Some early versons of the BASIC programming language carried the idea of an "imperative" language to its logical conclusion, and every statement in the language stared with a "verb". Arguably it's a bit more obvious what
LET A = A + B
means compared with just A = A + B, but since the computer doesn't need the "LET" to figure out what the statement means, it didn't survive into any "modern" programming language that I know of.
The inventors of COBOL language (now pretty much obsolete) chose not to use symbols at all, and went for long winded alternatives like
ADD B TO A
ADD B AND C GIVING A
instead of a = a + b and a = b + c. That might not seem too bad, but typing words like MULTIPLY and DIVIDE in full instead of * and / wasn't much fun.