The hiring bar at these places is pretty astronomical. Realistically it’s going to be well out of reach for a majority of even the very best programmers in the world.
Don't read too much into it. As an entry level applicant you're only getting a small fraction of anyone's attention and will need to apply Hanlon's razor to interpret most of your interactions.
(I'm sure they are some toxic interviewers and dysfunctional companies where that's not the case, but you don't want to work there anyway so it doesn't matter.)
A lot of good things flow from there.
First, you should feel a little more relaxed. This generally helps think more clearly, talk more naturally, avoid panicking, etc.
Second, you should shift your mindset to view your relationship with the interviewer as based on cooperation, not antagonism. This generally helps act more friendly and less defensive.
Third, if you were at home cooperating with a friend to solve a difficult problem, would you stay silent, or would you run your ideas off each other? Probably the latter. The same applies here. Unlike your friend, the interviewer is limited in how much information they can give. But again unlike your friend, the interviewer already knows exactly how to solve the problem! So leverage that. Ask them whether your proposed solution makes sense. Ask them whether the time and space complexities of your proposed solution are good enough or if you should keep digging. By the time you start implementing the solution in code, you should be fairly confident that you're going in the right direction. (Sometimes the interviewer won't give much away. Don't panic, it happens—in that case just go for it and hope for the best.)
where can one (I) test to find his position on it ?
I was super lucky to work with a gem of a recruiter at one of the FANGs who spontaneously pushed for 6 after my feedback from 5 came back positive. The other FANG was more reactive and didn't invite me back until I told them of the competing offer. So perhaps try pushing for it yourself if your recruiter is not as proactive and you feel you have a shot. Worst case they'll say no. (Or they'll say yes and you'll bomb the next round. That's not supposed to reflect negatively on your first round, although it would likely have affected me psychologically and made me less confident during the comp negotiation.)
I ended up joining a company without levels, so I'm afraid I don't have any more useful information for you than what I gleaned as a candidate.
Source: interviewed with 3 FANGs a few months ago without any significant open-source contributions, got L6 or equivalent offers from all 3. I also don't have a CS degree, for anyone who would feel discouraged by that.
Edit: I don't want to minimize the effort involved, though—I did spend a lot of time learning algorithms and data structures over the years, but I enjoy learning this stuff and find it far more relevant in practice than most people here appear to think.