Is this a bug?
This is no bug - the user is unregistered
Clicking on his profile shows that he is unregistered. Therefore each question he asks is handled differently. The handle shows that this is very likely the same person. Moderators have tools to check this and be even more sure.
When encountering unregistered users you should point them to appropriate resources on Meta StackExchange such as Why should I register my account? where the main advantages are explained as well as how to merge their accounts. Basically they have to send a form to StackExchange so that an employee can merge their accounts. This is nothing a moderator can do.
I normally try to tell them that it's easier to keep track of their questions and answers and they can start collecting reputation so that they get more privileges, such as voting and commenting.
Sometimes these people register - sometimes they don't.
Usernames are not unique
The username can be used as often as people have the same idea for a username. The important thing is the email-address (or the IP address if no email-address is provided) that is used to login. To cite Separatrix comment under this question:
We have 15 'Tim's 18 'Dave's with 0 rep between them along with 21 'Joe's most of whom do have rep.
For example there are 9 users with the exact username "Aaron". This one for example.
The reason why the picture shown in the OP is special and I first answered with the point about being unregistered is that we see the same Identicon, also called a gravatar. The Identicon is basically a sort of visual unique identifier based on the email-address used. That's why you see it twice there - he asked two questions with the same email-address (or IP-address), but didn't register, which is why they both are handled as if a different person asked them. The Identicon is a pretty good indicator if you are looking for someone and he doesn't change his profile picture. Normally people with the same username will have a different gravatar so that you can distinguish them.
You can learn more about Identicons on Meta Stackexchange: