Is it secure to use the same IV for encrypting a unique private keys? I want to store private keys in the database. I will never store the same private keys twice. As I understand the only reasons behind using a unique IV for each encryption, is that attacker can't see that you have the same data twice.

My server is generating the private keys. My server ensures that each private key will be only once in the database. If so, should I use an IV?
If you are encrypting keys, then you should use a key wrapping method. The SIV standard is the best for this.

Regarding doing something ad-hoc and without an IV, it really depends on the mode of operation. But, I always recommend against doing something non-standard.