I've been told that writing $x\equiv a,b,c \pmod d$ is abuse of notation, is it really? I've been told that writing $x\equiv a,b,c \pmod d$ is abuse of notation, and that I should always write:
$$
x\equiv a\pmod {d}\text{ or }x\equiv b\pmod {d}\text{ or }x\equiv c\pmod {d}
$$
How true is this?

I agree that it's bad notation to write it that way.  I would read what you wrote as asserting that all three statements below hold, so it's somewhat ambiguous and nonstandard at best.  Putting the word "or" in the equation as pjs36 suggests is better, but it's kind of strange to have a word in the middle of an expression like that.
You could write
$$(x \bmod d) \in \{a, b, c\}$$
if $a$, $b$ and $c$ have already been reduced modulo $d$, however.
