jovan has it right.
The depictions you have seen are mythical beasts. Although it is true that there are animals called dolphins, there was a time when few people would ever see one or know what they might look like.
You are cursed by your superior knowledge. If you are presented with a carved dolphin on a piece of furniture, you naturally expect it to look like the familiar (to you) dolphin we all know and love. In times past however, this would not be true. Unless you were a sailor, you would never have the slightest idea what this fantastic beast could possibly look like.
Thus the “dolphin” entered into the catalog of design motifs in the category of “Mythical Beast”. It’s an attractive motif by the way, and it’s use continued long after the depiction was known to look unrealistic.
Links to stylized dolphins:
These days some cabinetmakers are incorporating realistic dolphins into their work. Links below:
It’s sort of funny, but I think you could make the argument that these are not authentic dolphins. The reference to “dolphins”, or “carved dolphins” on a piece of furniture implies the familiar stylized depiction. That point is arguable, but talk to an antique dealer, mention dolphins, show them a picture of the two modern ones I linked to, and I think you’ll hear them say something like, “Oh! You mean those kind of dolphins.”
Oriental motifs too, are common in european furniture. The ball and claw foot is an obvious example. I like jovan’s link, because it suggests the possibility of this connection between the motif and it’s (presumed) oriental origins. It is possible.
Best type of furniture to see dolphins:
Early Georgian Pier tables often have them
renaisance revival styles