What type of hinge is this?

some sort of concealed hinge (not visible when the door is closed, so "concealed").  If you want to keep the same appearance, you need to get the same hinge type, and presumably, size.  They also tend to have a max opening angle, and that is actually one of the main purposes for that hinge type, to limit how far the door can be opened.

They are readily available through pretty much any hardware supply.  Just a matter of figuring out which one to get.

Take the picture to the hardware store. It's a common cabinet hinge.