cold storage is a device that is not connected to the internet (unless you plug it into a computer to access the wallet), so I don't believe the blue wallet vault counts as that.

Blue wallet is definitely better than leaving your BTC on an exchange, but blue wallet on your phone would be considered a “hot wallet”, since your phone is connected to the net. If you want true cold storage you need a hardware signing device like a cold card or bitbox02 and then use a desktop wallet like Sparrow. Bonus points for running a node!