So I'm figuring any sneaker is the [blank] shoe for its intended activity. Hence tennis shoes, or deck shoes
(I guess Vans Authentics were originally for sailing?) Eventually, maybe a 1980s basketball shoe explosion overwhelmed common understanding. For casual sneaker wear, people really only wear basketball shoes, skate shoes or runners. Why mark that your Jordan 1s as "basketball shoes" and your Stan Smiths as "tennis shoes" in your language when "sneaker" is much simpler, and acknowledges that both shoes coexist in a casual wear category.
Anyway, skate shoe has stuck around. Probably the influence of Vans brand identity paired with the power of skater-facing marketing. Half of "skate shoes" out today function exactly the same as anything on a marshall's rack, beyond maybe a foam insole and also, all that awful but useful rubber paneling under the uppers. A skate shoe is coveted, argued about - it is not in the rhetorical section a baseball cleat or squash shoe might be. A skate shoe is attached to a professional athlete and comes in a variety of colors and styles. It is both a casual sneaker and a functional one. But nobody says "basketball shoe," which exists in the same realm. NBA pros still play in the Jordan 1 sometimes (well, in Off-White colorways.) This is weird.