I was reading articles on 13th depository and wot.wikia, and both sites state that Bethamin meant "heart friends" to be a sexual thing.

The quote from the series is:
Bethamin ... wrote a suggestion that Zushi be moved from the Palace to somewhere she could be kept in a double kennel with a damane from the Empire, preferably one experienced in becoming heart-friends with newly collared damane. Sooner or later, that always put an end to tears.

I find it pretty damn unlikely that the sul'dam would train a damane to flirt with other damanes, certain that a forced sexual relationship to a woman is just the thing to soothe a fearful young girl to accept her slavery. IMO, it just means a regular BFF (even the double kennel isn't that weird, people do have sleepovers after all). And though it doesn't really prove much, Alivia had clearly learnt nothing about romantic relationships in her 400 years as a damane.

Coincidentally, in my native language "heart friend" is a real term to describe a good friend.