I have tried stand up paddleboarding before, and the instructor encouraged us to wear our normal clothes, rather than change into wetsuits - "what you're wearing will be fine". Needless to say, we got soaked and the English Channel wasn't too warm that day!
It seems that an indoor variant called Floatfit has developed; essentially a paddleboard workout. From what I can see, participants are encouraged to wear sportswear, rather than swimwear. Some of the sites are specific about what should or shouldn't be worn. There are some photos and videos in the links below.
I think this translates to "Wear tight sports clothes that will be comfortable when wet", although my German is a little rusty!
https://www.baederland.de/aktuelles/aqua-floatfit-100098.html
Not sure how accurate this is, but Google translated this is "It is important that the clothing should be wet and not heavy"
http://malkander.nl/aquasporten/aquasporten/floatfit
"What to wear; fitness kit that you do not mind getting wet (swimming costumes/speedos are not suitable)"
http://charterhouseclub.com/2016CHnew/floatfit-coming-soon/
https://www.edinburghleisure.co.uk/news/floatfit
https://www.girltweetsworld.com/the-most-fun-you-can-have-working-out-caribbean-floatfit/
http://www.dinnerstories.co.uk/2016/10/floatfit-hiit-with-sport-aberdeen.html
https://www.telegraph.co.uk/health-fitness/body/floatfit-water-workout-making-splash/
https://www.standard.co.uk/lifestyle/health/aquaphysicals-floatfit-class-hiit-on-water-at-haymarket-hotel-review-a3299431.html
https://aquaphysical.com/class/floatfit-hiit-haymarket-hotel-london-4/
https://onin.london/floatfit-review/
https://www.facebook.com/pg/FloatDXB/photos/?tab=album&album_id=431165450569555
https://www.youtube.com/watch?v=FxVSeczz3nE
https://www.youtube.com/watch?v=-tgI9GD4R2Q
https://www.youtube.com/watch?v=jHXZrhI57fo (from 1:29)
https://www.youtube.com/watch?v=fSKHZ77HE2o
https://www.youtube.com/watch?v=rBW54s_txSY
https://www.youtube.com/watch?v=gA79MJOk5GQ
https://www.youtube.com/watch?v=8o1i7ZZ4RNU
https://www.youtube.com/watch?v=NqRptOTwefI
https://www.youtube.com/watch?v=CgA5lPF3EGA
https://www.youtube.com/watch?v=OQx9awrW_-M
https://www.youtube.com/watch?v=-oDjvRydq8s
https://www.youtube.com/watch?v=Qx1gXyK_OOA
Looks fun!