“I told everyone and their mother about how I won the contest.”
“Everyone and their mother” meaning that I told many many people.
Would this be considered slang or just a figure of speech?
It’s a figure of speech, the more traditional version being (as Kate Bunting has commented) ‘[All] the world and his wife’.
To use the sentence literally:
“I told everyone and their mothers about how I won the contest.” would suggest that you told all the other competitors, their mothers and anyone who stands still for long enough.