Weakening of Paillier cryptosystem due to ciphertext equivalence and order in CryptDB
Your hunch is wrong because of the definition of CPA security: Assume that some knowing some kind of relation between two plaintexts would give the attacker an advantage. 

Now think of the INC-CPA game: Nothing stops the attacker from choosing exactly this kind of relationship. And if the scheme is IND-CPA secure, knowledge of such a relation does not break the scheme. As mikeazo pointed out: Even if you only use a message space of $\{0,1\}$, CPA security still ensures the attacker can not distinguish their encryptions.

The other aspect you hinted at: Under CPA security it is not possible to distinguish whether two ciphertexts encrypt the same plaintext: The attacker can encrypt $m_1$ and $m_2$ as often as he wishes. If there was a way to determine that one of the encryptions encrypts the same ciphertext as the challenge, he wins.