Oh dear. You seem to think that Bitcoin is some website where you stick your private key in and get authenticated as the "as the owner of that wallet".
Can you use the API key to control your funds without using the website that issued the key? Does the API key allow you to receive/spend funds offline (cold wallet)? Can you use the API key to sign a transaction and put it on a public ledger where anyone can verify it?
Just a few basic features of a Bitcoin private key off the top of my head. I'm sure you can find more on Wikipedia.
What you do after the being authenticated depends on what service/tool you are using.
Of course a private key let you do different things than an api key of an exchange, but some of them are in common, that's why i keep saying that can act as.
Can you use the API key to control your funds without using the website that issued the key?
BLOODY YES ! That's the fucking point you don't get. you can access every function of a service via any computer terminal without the need of using the website. What kind of functions you can access is based on the api key privileges.