Yes, it is the same concept since the fibre products of schemes DO coincide with the fibre product of the underlying topological spaces in the case of open immersions (Check, for instance, Hartshornes proof for the existence of fibre products in the category of schemes).
As for the second question, I don't know how irreducibility of algebraic spaces is usually defined, or if the concept is important.
One way would be to define it as irreducibility of the underlying topological space. Look in section II.6 of Knutson to see how this is defined. He topologises the set of underlying points (equivalene classes of monomorphic K-points) by using closed immersions. Check if this is the same as topologising using open immersions (this is probably easy). If it is, my guess is that irreducibility of the underlying topological space is equivalent to your suggestion.