A simple search and test would give you an answer. Plus we in the Bitcoin community follow a Golden Rule - Don't Trust, Verify.
But because I was also curious, I tried the wallet and I have verified for MYSELF that the wallet doesn't actually need a Coinbase account. It's a wallet with your own keys that could receive Bitcoin, and all sorts of shitcoins. Because it's a browser extension, I advice using a Ledger hardware wallet. It's the only hardware wallet allowed to connect with Coinbase wallet.
Sure, I could just download it, but since I was not going to use it, it wasn't necessary so I just asked. Since you mentioned "trust", I don't trust anything with a "coinbase" in the name, apart from the coinbase transactions in mining.
And I definitely will never use a Ledger wallet for a variety of reasons that have already been mentioned in the forum.
So, a wallet that I don't trust the name of and supports shitcoins, that only accepts Ledger as a companion hardware device. Instant "no" from me.