I'm the professor

[Ostap]

I hear you all ask 'bout the meaning of scat
Well, I'm the professor, and all I can tell you is
While you're still sleepin', the saints are still weepin' 'cause
Things you call dead haven't yet had the chance to be born
I'm the Scatman

Hello, teachers. Another line from the song Scatman, where I think "I'm a professor" would fit grammatically, so what is the reason for using "the"?

 

[Skrej]

He's the professor of this specific class.*

*apparently 'Scat 101'.

 

[probus]

And by the way, scat is another word for sh*t, in this case probably of the bull variety.

 

[Glizdka]

It's not about what is grammatically correct and possible, but what is meant by the speaker and wanted to be conveyed.

"I'm a professor" tells us what their profession is, in general. They're not a firefighter, a nurse, or a cheerleader; they're a professor.
"I'm the professor" tells us what their position in this class is, specifically. They're not one of the students, apprentices, or auditors; he's the professor.

It's somewhat meant to show that, out of all the people in class, they are the one supposed to know what scat is.

 

[probus]

I agree with @Glizdka. "I'm the professor" is equivalent to saying "I know all about this" or "I know far more about this than anyone else here."