Airdrop Name is Incorrect
Step one on the Mac might be to sign out of iCloud. AirDrop uses that.

Before doing anything, make a full backup of the Mac. Then check for other user accounts on the Mac in system preferences. Once you're sure your account is the one doing the naming for airdrop, you can see if your iCloud contact details are correct and sign out (deleting any iCloud data on the Mac when prompted since that data is in the cloud) and then sign back in to iCloud.

Worst case, once you've singed back in, check the Contacts app and check that "your card" is really your information. Search for "My Card" in contacts app help if you aren't sure how to find that menu option.