It's pretty easy to identify who is who. All of the skins have pretty consistent silhouettes because they always have their gadgets attached to them somewhere or wear a particular type of outfit. Sundance is always in a jumpsuit and has a rig with grenades hanging off it, Dozer is always in EOD gear with a neck blast plate, Mackay always wears a flightsuit with a harness and carabiners, Zain has prosthetic legs and the XM370 on his back, Boris has the turret, Lis has her missile launcher, Crawford and Irish have backpacks in different colors, Falck has the big medkit on her back, Angel has the duffel bag/Pelican, Casper has the drone on his back, Rao has the ECW suite, Paik has the visor, Blasco has the jammer on her back. Every specialist has unique animations in the pre-match screen and also in game, too, i.e. Falck always stands up straight holding her gun in one hand pointing upwards, Casper always "sneaks up" and stands slightly hunched with his gun at low ready position, etc.
The animal skull helmet is on last season's tier 100 battlepass skin for Dozer called Overthrower. It doesn't show up in the game store or in his available skins because it is no longer available to obtain since it was a battlepass exclusive. Previous battlepass skins only show up if you have obtained them.
I do wish we had an option to select a "generic" soldier like in BF3/4 though. The whole thing with a bunch of identical "specialists" running around is a bit goofy. Even moreso when a completely generic and unidentifiable soldier is more unique than them since it could be literally anyone under the balaclava instead of knowing for a fact that the multiple people next to you are meant to be a specific person.
And to answer your question, you can't turn off skins.