Learn English – Which is grammatically correct: ‘update on’ or ‘update about’ the status

They are both grammatically correct. However, it is much more common to see "update on" rather than "update about". I would also say they mean slightly different things:

I will update you on my status.

Means "I will tell you what my status is." It is very clear and definite.

I will update you about my status.

Means something like "I will describe my status to you." The word about evolved from the figurative use of a word meaning around. This can still be seen by uses of about that mean approximately (e.g. "It's about a mile"), and there are even still uses that mean physically around (e.g. "Rotated about an axis"). Speaking "about" something implies that you are mentioning the qualities of the thing and discussing related subjects, not simply saying what the thing is.