My mother has my brother living with her since he lost his job 5 years ago.. he pays no rent and has no agreement to do so. The arrangement is open ended.

18 months ago my bother invited his homeless friend to stay for "a week." He is still there and camps in the backyard.

I think the friend is considered a lodger. But is my brother as well? It seems simple to evict a lodger if you have a single lodger, but what if you have two? How do the rules change?

The rules don't change for one or two. You have to give each one OR which ever one you want evicted a xx days (depending on where you live) notice to quit . Then take them to court where they will be served with xx days to vacate. Then get the sheriff or constable to escort them out if they have not left on said date.

No. The rules are all different if the state you are in does in fact differentiate between lessees/ tenants and "lodgers". A few do. What state (or country) is this?

I know, in CA, a lodger refers to a single person. If there are more the rules change.

On the other hand, since the brother is a family member, he could be part of the family household.

Also, someone camping out (I assume in a tent) in a backyard, doesn't seem like either. Has he been asked to leave?