Just for the record, what y’all are talking about are, in general, the critters that are depicted holding up the shield, which are referred to as supporters, the shield being an escutcheon, and the whole depiction, with shield, crest, supporters, motto, and miscellaneous other details, is an accomplishment (the objective name of the collection of heraldic gadgetry, not an abstract-noun critique of the abilities of the owner thereof).
Though it’s getting downright pedantic to insist on the distinction any more, a “coat of arms” is just what it sounds like – the tabard, an item of overwear, sleeveless, the design of which depicts one’s blazon (the official legal design of the escutcheon) – or, more usually, the blazon of the person employing those who wear the tabards.
It’s quite possible to have animals on the escutcheon proper, e.g., both England and Scotland use lions, the English arms having three lions passant gardant (walking sideways and looking at the person observing the shield), while the Scottish lion is rampant (rearing up). But while the pre-1603 English shield was supported by lions, the Scottish shield was supported by unicorns, and when Great Britain was formed as a political entity, the two shields were “quartered” (draw a shield and line in a cross-pattern, place one set of arms in positions 1 and 4, and the other in positions 2 and 3 on it), and one supporter from each set was used. (This omits the display of the French “arms in pretence” and the use of the Irish harp since 1800, for simplicity’s sake.)
To the extent that the unicorn represented anything more than purely imagination, it appears to have been a rare sport among aurochs, in which the horn cores merged, and the resultant male displayed a single horn and was powerfully but slenderly built (apparently the result of the merged horn cores’ pressure on the fetal brain and pituitary gland) and exhibited a rather take-charge air when full grown. It was not, however, a mutation which bred true, but a rare congenital condition. So, for what it’s worth, a unicorn would taste like beef.