What are the dangers of predictable (repeated) plaintext structures?
Well, no, modern ciphers (such as AES) do not share that weakness; they are designed specifically to be secure without any assumption of what the plaintext looks like.  You can encrypt the same message with multiple keys; in fact, you can encrypt the same message a bunch of times with the same key, and the attacker still cannot deduce any information for what the plaintexts look like (or indeed verify that all those ciphertexts did in fact correspond to the same plaintext).

As for specific terminology for an attack that, given a large number of messages, attempts to find some information about them, I have seen the term "statistical cryptanalysis"; I don't know how commonly that is used.

As for resistance to AES with a secure mode to statistical cryptanalysis, one way to derive it is indeed based on ciphertext indistinguishability.  If the cipher mode does provide ciphertext indistinguishability (that is, the ciphertexts corresponding to chosen plaintexts are indistinguishable from random outputs), then any process that gains any information on the plaintexts based on the ciphertexts leads immediately to a distinguisher (as doing the same analysis on truely random outputs can't give any information about the plaintext), and such a distinguisher contradicts the assumption of ciphertext indistinguishability.