So what kind of ship is the Iowa?
Nothing else is said about it, aside from one line of dialogue from Kirk that he was born in space, aboard the USS Iowa.
How does the ship apparently look exactly like the prime universe model then? Is that explained?
Nothing is explained. La'an is on the Bridge of the Enterprise for a minute or two and the graphics in the background look just like Pike's ship. The main difference, besides Kirk being Captain and Ortegas being the navigator instead of at the helm, is the ship being designated the UEF Enterprise and the insignia being different.