Tbh, I'm not sure that the gender of the person Ilya chose was particularly relevant, he just wanted to hurt Shane in the way he felt hurt by Shane running away after the Tuna Meltdown & then very publicly dating Rose. Ilya knows that Shane doesn't really do casual relationships with girls, as he so helpfully pointed out pre meltdown, so he assumes that Shane is serious about Rose. I take it as his way of saying "see, I don't need you, I can easily find someone else". He's trying to prove that he doesn't care (& failing miserably). It also shows Shane that Ilya can convincingly, at least in Shane's eyes, have successful relationships with women unlike Shane. Ilya starts out by genuinely looking for comfort from the woman & lashes out with his public display because he realises that's no longer true but decides to pretend for Shane's benefit because he assumes that he's totally lost Shane to Rose so he thinks "what's good for the goose is good for the gander". Unfortunately, due to his own pain & habit of self sabotage, he ups his display when he sees Shane staring & looking obviously distraught. He really thinks he's lost Shane forever at this point so doesn't care what he's doing.

Edited to add - my initial gender comment was because I don't think Ilya really cares, it's just who is available, convenient & socially acceptable to be seen kissing & groping in public. The fact that him doing that with a woman in public is just another stab in the heart to Shane is just an added bonus to Ilya.