Do letters of recommendation typically include a biography of the writer?
Yes, this is relatively common. (In my opinion, not common enough.)

Recommendation letters carry more weight if he reader knows the writer's qualifications.  In particular, letters about qualification for graduate school carry more weight if the reader has some sense of the writer's track record for judging students' research ability.  (This is why letters from senior faculty are more valued — not because of their vast research experience, but because they've presumably seen and evaluated more students.)  The best way for the reader to understand your track record is to tell them.

But briefly, because as you suggest, the letter isn't about you.

I don't think there's a standard way to do this.  Some people put their bio early in the letter, say in the second paragraph (because the first paragraph is the actual recommendation).  I put mine in a footnote on the last page.