I have 2 children with live with both parents in the same house but we are not married. Can both of us claim a child

Each of you can claim 1 of the children if you want to and agree to. Or 1 parent can claim both children. It's up to you to decide. Each child can only be claimed by 1 parent.

No. Only one parent can claim the child. If you are asking can each of you claim one child- then, yes. One parent can claim one; the other parent can claim the other.