the website would have requested a public address, which is a key that can be shared freely anywhere without risk. the way this key is used is to send you dogecoin. no dogecoin can be withdrawn with that key and it cannot be used to make purchases.
When using a cryptocurrency (at least those based on bitcoin that I'm familiar with), you can only send dogecoin from your wallet with the private key. in the case of dogecoin core, the private key is stored in wallet.dat, and should NEVER be shared anywhere ever, you should guard your private key with your life, as anyone who has it essentially has ownership over your wallet. You can encrypt your private key so that transactions can only be made with the manual input of a password.
When sending cryptocurrency to another wallet, all you need to know is the recieving address (which is what this website would have asked for). I was about to say that the function of adding a wallet address to your account here would be for recieving dogecoin tips, but I can't actually see a tip address on any profiles, so I guess this would be a question for one of the guys who run the site. perhaps as a simple filter to weed out the most basic bots? since all you need is the most basic knowledge of installing a wallet and generating a recieving address. I'm not 100% sure though.
Regardless, there is no harm in sharing a recieving address, should look like
DMyuE5VpSQMdaKSgWbdcgofhHGUCDC8nkC
But if anyone ever asks for your private key or wallet.dat, they are trying to scam you and you should absolutely not share it.
