many a: distributive idiom

many a is a distributive rather than aggregate idiom

Many a is distributive because it refers to things individually, just like each, every and either.

All of the above expressions can be followed by "one", e.g.

Each one was a perfect sphere.

Every single one of them was drunk out of their minds.

When it comes to shoes, you can put either one on first.

There are lots of dogs in this area, in fact I've seen many a one without a collar or lead.