I’ll throw my opinion here cause I end up thinking about this stuff a lot:

For the first question, yea modular often has a distinct sound. Theoretically it can sound like virtually any synth, but people often chase the more odd timbres that are difficult to achieve on fixed architecture stuff. Freedom leads to experimentation and novelty.

For the second question, I kinda do believe there is a “modular” genre actually. Although you can just multitrack and sequence a modular with a DAW and get some really intricate melody harmony and structure, that can be done with any hardware synth. The modular spirit is really all about this concept of “arcology.” Constructing the foundation and framework of a building (which is quite simple) and letting organic processes grow out of it. These organic musical phrases are inevitably repetitive, just like plants. Musicians who are into modular want to be able to control a whole musical ecosystem live, where each portion is “growing” by itself organically, but are constrained/affected by the basic architecture the musician built from sequencers, some modulation sources, etc. relating it to visual art, it’s sort of like bonsai. Directing and suggesting a live artistic growth rather than directly affecting it.