Thought I better reply to this post quickly because I'm leaving the country for about a month or so. I should be back, but it may be too late.I've never heard of anything like this at all, at least not here in SF. One thing I've learned over the years that doesn't seem peculiar to any particular AMP is that AMP girls generally talk to one another about their customers. In your case, it could be that your info packet at this place is stuffed with stories about how you are potential boyfriend material. That could be good, if it leads to a couple of nice, unconditionally friendly dates. It could be bad, if it leads to an expectation from a girl carrying a temporary, soon to expire visa, that you are potential permanent visa material. In other words - marriage for convenience. Something to watch out for when girls are leaving the country. If they're here legally, it may be for just a short period of time. Depends on the visa.
Or it could simply be that you happened upon the great, previously undiscovered vortex in the spacial fabric of AMPs. That mysterious massage parlor were the only reason the girls are working there is because they have determined that's the best way to meet prospective boyfriends. We've all heard such a place exists (well, at least we all have dreamt that at some time or another) and you found it. And before you can say "When are you coming back to the country?" not one, not two, not three, but four potential girlfriends materialize.
Jesus, I always knew this place existed! I don't know why I didn't find it!
Seriously, if I were you, I do exactly what I felt comfortable with and nothing more. It seems like the more girls you see, the more complex the picture becomes, so you might "settle down" with just one (from that parlor) and see where that goes. In my experience, if you cross the romance line with any AMP girl, even slightly, and then see another girl at the same parlor (or in your case, the whole damn parlor) it's trouble.