Is zero IV with AES-CBC safe if CPA is not possible and indistinguishability is not an issue?
Furthermore, is using AES-CBC this way any safer than using AES-ECB?


Yes, using CBC mode is (almost?) always safer than using ECB mode.


  Is using zero IVs with AES-CBC safe[...]?


No it is not safe, because if you re-use a key with that construction you will leak the common prefix between messages, at block granularity. That is if you have three messages encrypted with CBC with the same IV and key $m=A\parallel B\parallel C\parallel G$ and $m'=A\parallel B\parallel D\parallel G$ and $m''=A\parallel E\parallel F$, with $A,B,C,D,E,F,G$ being 16-byte blocks, you can tell that $m'',m'$ and $m$ share the first block in common and you can tell that $m'$ and $m$ share the first two blocks in common, of course you don't actually directly learn the value of these shared blocks.

Note that either changing the key or the IV for each message fixes the above problem.


  There are more modern modes (which are authenticated) which only leak equality of messages upon a (IV,Key) re-use: That's kinda what I'm
  looking for. Which modes are those?


These modes are called nonce-misuse resistant authenticated encryption modes (nmrAE). The two standard suggestions for this are AES-SIV and AES-GCM-SIV or AES-PMAC-SIV, both of which will offer standard authenticated encryption security if you use different (IV,Key) pairs with each message and will both offer (some) security if such a pair is re-used and of course they offer authentication as well and only leak equality of the underlying messages upon a re-use.