When a blockchain hardforks, it basically is like doing File>copy File>Paste, so now there are two identical blockchains, but they won't continue to be identical from that point forward. If you control your private keys, you control the bitcoin for those private keys on both chains.
What coinbase is saying is that if a hard fork occurs, you as a customer of theirs, will not have access to both chains, like you should as a bitcoin holder. Instead you will have access only to whichever chain they decide you will have access too. It is a scam by coinbase in my humble opinion. A scam which results in coinbase controlling every single coin on one of the chains, and their customers retaining the coins on the other chain.
The coins don't just "vanish". If coinbase has access to your private keys, they have access to both chains, no "support" is necessary. They are using jargon and rhetoric to manipulate and confuse people.