I never understood that terminology since the middleware was always on the server too.

It can be on the client side. Middleware just means something between the operations of the application on a server and operations of the application on a client. Doesn't matter where it sits, it's just the link.

But to the parent comment, yes, I agree, this is a middleware.

I thought of middleware as sitting on one server, with the database behind it typically on another.

Yea, I've seen that often. Some people would call any server side code middleware.