I have amateur videos downloaded 20 years ago from Messy Fun which feature women covering each other in emulsion paint from the 90s, who as far as I know came to no harm. These days all paint comes with health warnings like it's radioactive and instantly toxic to all living things within 200 miles but in reality emulsion paint should be fine, as long as exposure isn't too long and you keep it out of eyes and mouth, and clothed is probably safer than nude. I'd avoid any form of gloss though.
It used to be a case of water based architectural paint was okay - avoid solvent based.
It's slightly more complicated now since the VOC regulations have meant that many more coatings are water based.
If I use emulsion I use 'contract' matt (there are many brands like Dulux 'Supercover') which is basically a paint for new plaster. It's opaque but, crucially, it never sets or polymerises like acrylic or vinyl so it will break down under stress or warm water.
The paint in your link would be okay in theory as long as you don't let it dry anywhere.
Basically make a note to avoid the harder-wearing-when-dry versions of water based paint.
This is a good explanation of the differences for ref.
You must avoid getting emulsion in eyes or mouth. In theory the emulsifier should react with oil in skin so should be used with caution. That, of course varies from brand to brand. As with any substance new to you, do a test on a small area of skin. That includes different brands of the same nominal substance. All that said, the texture and coverage are great.
I've used both white household emulsion and school poster paint without any problems, despite pretty heavy coverage. Got a bit of emulsion in my eyes once or twice and it stings a bit so I stopped and washed it out. No lasting effects. Paint fight sounds fun but not quite as safe for eyes as pouring / wrestling?
I've used regular acrylic non toxic craft paint for alot of years sex included with no problems. Not to say you couldn't have any. You do have to make sure it doesn't dry a whole lot in hair. And will stain so I'd be careful. But it's the best coverage I feel without streaking