Usually we still ask for statistics to be at least valid (i.e. have a significant signal under a null hypothesis). This paper doesn't even do that. It's like claiming no humans have been to the moon and then "verifying" this by randomly asking a million random strangers on the street if they've been there.

I think they are saying that there is no proof of being injective. The argument with the hash is essentially saying, doing the same experiment with a hash would yield a similar result, yet hash function are not injective by definition. So from this experimental result you cannot conclude language models are injective.

That's not really formally true, there are so called perfect hash functions that are injective over a certain domain, but in most parlance hashing is not considered injective.