Father has 2 year old child, hasn't seen biomother for awhile, but has been staying with father girlfriend. Girlfriend wants father to give her joint custody of the child. Can this happen with or without the biomother

Nope. Not going to happen. If she is looking to adopt the child your going to have to be married first for at least a year to show stability.

The girlfriend has no legal standing for the child. And in the case of the adoption route you will need the bio-mothers permission.

Only a court can grant or set custody. And I doubt if a court would grant custody to a girlfriend. Instead they will suggest adoption which means qualifying and getting the mother's assent.

This is not possible. The girlfriend is just that a girlfriend and has no legal standing regarding custody of the child. Only when the girlfriend becomes a step mother then she would have standing in the matter.

... girlfriend wants father to give her joint custody of the child. ...

What is it you suppose that would mean?

Joint custody - with the father being the joint custodian with the girlfriend? As opposed to the mother? The GF can be delegated the father's rights - as opposed to the mother. Same way as a school or babysitter can be delegated a child custodial parent's custodial rights.

But what the father gives can be taken back. Suppose, for example, the father and the girlfriend break up. The father would be able to say that she no longer has any rights.

The only meaningful "joint custody" would be ordered by the court, and it isn't likely that a court would give the girlfriend such rights.