Exactly. It's a specific type of seat, designed for a specific purpose, so we refer to it using it's specific formal title.

Bar stool
Pew
Throne
Deck chair
Mission chair
Wheel chair
Recliner
Beach chair
Chaise lounge
Ski lift
Captain's chair
Park bench
Cleopatra chair
Adirondack
Balans chair
Director's chair
Morris chair

Etc..........

And on bicycles, we find SADDLES.