Is a family crest a symbol?

There is no such thing as a family crest; crests are part of the overall heraldry held by an individual (usually a man). They refer to the crest worn on top of a knight's helmet during the late medieval period - not to the design on the shield.

A crest can be many kinds of things like a swan, a deer's head and neck, a man's arm holding a sWord, a ship, a standing dragon and so on. This crest was often used as a badge by the knight's retainers and servants, who were not permitted to have a coat of arms.

Fraudsters have in modern times pretended to be able to identify "your family coat of arms" in return for payment; this idea is completely false because if you have been awarded a complete coat of arms (with motto, supporters, mantling and crest) you will already have a legal heraldic document proving it - nobody without that evidence can claim to have any heraldry.