The free play styles of Genos and Genos2 are
80sChilloutSynth in Retropop
ElecOrganBallad and
TheaterOrganBallad in Easy Listening
EtherealMovie
EtheralVoices
MovieHorns
OnBroadway and
RomanticMovie in Movie&Show
AmbientChristmas and
OrganHymn in Classic & Events
Brassband in Trad & Folk
It is no way to find them automatic. They are only recognized by the word Free Play onscreen. I checked the channels on one of them, and the Rythm1 and Rythm2 are active. No beat, but it can be a hi-hat or some other atmosphaeric instrument that is used. A tempo is given for each style, although you are free to play in your own tempo, speed up or slow down.
The windows program "Genos2 StyleFinder" from Oostendorp Muziek (free) show you all styles on Genos and Genos2 with search functions for DJ, Free Play and Session styles, and for time signatures and tempo ranges. That is very useful for finding styles. For me it would be more useful if it was an Android app on my tablet. See https://forum.psrtutorial.com/index.php?topic=68711.0
Something similar should be possible on Styleplayer.io. We already have Time signatures and Tempo, but they are a part of the Song page, not the Style page. Styles have a standard tempo, the song where it is used may have a different tempo. The time signature will be the same for both. There could be a dropdown or a free text field on the Styles page. You have already on your todo-list search functions that could be used to find styles.
I don't know how frequent Free play styles are outside of Genos keyboards. They are very nice to play for certain musical styles. They probably must be found and registered manually.