Would I be able to set up two Apple ID's under my name?

Yes, you can do this, but you should consider exactly what you’re trying to accomplish and whether this is the best method.

Each Apple ID looks like a completely different person.

For example, one reason you might consider doing this is you have two very different identities that you choose to use - one for your personal activities and a different one for your professional or business identity.

Matvic12 wrote:

I thought the second one would be like an alias to the first one as there would be 2 ids with the same name and info in the system?

Absolutely not! Two Apple IDs look like two distinct people.

I still don’t understand what you’re trying to accomplish. But perhaps it can be done using different email addresses on the same ID. If so, and if an iPhone or iPad is involved, go to Settings > [your Apple ID] > Sign-In & Security and add an email address at the top.

“Start fresh”? As in permanently and forever remove all previous data associated with you in Apple’s ecosystem? You may do that using your current Apple ID by simply setting up your new iPhone as a brand new device, never importing anything related to previous history (such as an iOS backup or QuickStart from the previous iPhone).

Some iCloud data could be migrated. For example, an iCloud contact could be shared. An iCloud Photo could be shared using the Share button.