Some 240V rated equipment will not be happy on 208V. A hearing test trailer should have 120V equipment for the testing, so whether it's fed from a 120/208 or 120/240V source *shouldnt* matter.
As to your second question, residential service is often single phase (split phase or center tap neutral), with 120V to neutral but 240V line to line. Commercial transformers are often 3 phase and due to phase angle, 120V to neutral gives 208V line to line. There are open delta transformer banks that have 240V line to line, but the B phase is 208V to neutral. Why have 120, 208, 240, 277, 480V here? Different demands of equipment is one answer. A 900HP aeration blower would be wired to 4160V, you could wire it 480V or even 120V if you could make conductors that would carry 4,000+A. Cheaper to make better insulation and smaller wire.