The story
The nickname comes from their resemblance to dogs in various respects, which is largely explainable by their phylogenetic relation to dogs; and the fact that they live in the sea.
Etymology from Wiktionary, CC BY-SA 4.0.
sea puppy
Today it means: A seal, especially one of the cutest and furriest kinds of seal, such as a fur seal or a sea lion, and most especially a juvenile thereof. Full definition →