Basement Recreation Room
Yes, your two exits are code compliant for a basement rec. room.  (In fact, only one exit is required. It can be directly to the exterior or up to the main floor.)
Two exits are not required unless you’re going to sleep in that room.
The Code only stipulates the size of one door in a residence. It requires one egress door that is (1) 32” clear opening with the door in the 90 degree position (which requires a 36” wide door), (2) located at a exterior entrance into the residence opening into a public way or open side yard that opens to a public way, and (3) cannot open into a garage or carport, (4) and shall be a minimum of 78” high measured from top of threshold to bottom of stop, (5) and must be side hinged...no sliding doors allowed. (See 2017 ICC R311.1 and R311.2.)
Don’t forget the ceiling height shall be a minimum of 7’ too.