Q:

Having/being a sweet tooth

Several times I came across the expression "I am a sweet tooth", not "I have a sweet tooth".
Is it allowable in colloquival speech?
Example 1
Example 2
Example 3

A:

It's not natural English, but that doesn't mean you couldn't use it for humorous effect.  This relates to the concept of metaphor, where you relate the characteristics of two objects.

He is a real mule when he doesn't get his way.  He digs in his heels and no one can move him.
The University is a beacon of hope for those who aspire to an advanced degree but lack sufficient funds, since it doesn't charge any tuition and provides a living stipend for underprivileged students.

In the same way you could have the following conversation:

A. I confess, I have a serious sweet tooth.
  B. That's nothing. I am a sweet tooth!

The suggestion here is that you have such a weakness for sweet foods that you can be described as a "sweet tooth".  But, again, this is just a clever metaphor, and not a common idiom.

