If I were to have more than one email account with Gmail, is it necessary for me to have the same user name?

In other words, with “exampleaccount1@gmail.com”, my user name may appear as Hawaii Jake. With “exampleaccount2@gmail.com”, could my user name be John Q. Public?

Would there be any bleeding between the two accounts that might cause servers to get confused?

I already have one account for everyday use, but I want an anonymous one to use for various purposes. You can get your mind out of the gutter.