I'm not an ampersand person is a very elliptical statement.
When someone likes dogs, or is a so-called "dog-lover", they're called a dog person, or if they like cats, a cat person.
Do you like dogs?
-- No, I'm more of a fish person.
So what that sentence means literally is "I am not a lover of ampersands". "I don't like ampersands". But figuratively, what the person in that video is saying is "I don't think of myself as half of a duo, and I don't want people to think of me that way."
Here's how that meaning is arrived at...
When two people become known as a duo, especially in popular entertainment, they often get referred to by a "commercial name" that takes this pattern:
name1 & name2
It is a kind of "billing", how the name would be represented on a poster advertising a concert, say.
For example, these names from American popular culture from decades past:
Sonny & Cher
Captain & Tenille
Simon & Garfunkel
You can see this tendency to use the ampersand with duos at this site.