Bitcoin only wallets maybe, others not. In most cases it must be connected to a device that is online and other software.
Many hardware wallet support multi coins. So I really don't get what your fuss is all about. It's no longer a cold storage if it has any bit of it connected to an internet. Do you know how an airgapped device is created? No it is not. For a wallet to be an offline wallet it must be offline all the time. It can't be offline some of the time. If it could, then where do you draw the line? 10% offline time is enough? 20%? 50%? Either it is an offline or online wallet it is a binary choice.
I guess there's a mix up here. It's a cold wallet as long as it remains offline and has no mean what so ever connecting it to a network. It's not called an offline wallet, but a cold storage. Quote
Nope that is not correct. We have clarified here. You connect a closed source device to another that is connected to the internet. That can never be a cold wallet. It could be doing anything while you are not even touching it. I say that cold wallets are as close to airgapped wallets as possible, others here want them to be as close to extension wallets as possible. 
A lot of people already gave the correct explanation about this mate. You don't connect your cold storage to the internet, you only use them to sign a transaction offline and take the signed transaction to a watch only wallet which it will be broadcasted to the network.