Are you saying that the only consideration for the Prof to accept you (beyond academic
qualifications) is the funding ??
But, I think, 95% of the admitted graduate students are funded, so they are all equal
with this regard, aren't they?
95% of admitted Phd students maybe, but not MS students which is what you're talking about.
Even for the Phd program most places only guarantee funding for the first year, sometimes second year. After that you are expected to find a research group, which is who funds you. So if they're low on funding you might not get in. The issue you're dealing with, in the MS situation, is that they may have funding but they're reserving it for Phd students.
Also, money is far from "the only consideration", but without it you can't even begin to entertain the idea of taking on students who require funding. Therefore, it is effectively the most important aspect.
1) Can that scenario really happens?
2) If yes, then what should you do then?
I would assume that it could happen, rarely, and maybe even has. Like said above, you need to check these things before you enter and be aware of the financials while you're there.
Imagining being in that situation I would see only two options, assuming these two profs are the only ones in your field of interest. One, find another field with another prof, though I wouldn't do this in a Phd program but maybe in a MS program. Two, leave the school, with their blessing if you can, which if the funding is really that bad they probably will understand. However, like I said, this being an MS situation funding isn't really expected so you kind of have to deal with it. If funding isn't the issue, ask these profs what it is, that should give you real insight into the problem.