Part of the problem here is that we treat true/false as "just another value" in programming, and thus the usual operators are used to compare them. In math notation, if you wanted to compare the result of comparison A=B to a Boolean value C, you'd normally write something like (A=B)≡C, using different operators for equality and equivalence.
Interestingly, programming languages haven't always treated Booleans in this way. If you look at ALGOL 60, it starts by cleanly separating expressions into arithmetic and logical ones, and defines completely different sets of operators for each (the reference language also uses = for numeric comparisons and ≡ for Booleans; and boolean inequality is, of course, just XOR).