Is the phrase
[all] up in $POSSESSIVE_PRONOUN grill
which is synonymous with the figure of speech
in one's face
an automotive metaphor?
If so, would it be more correct to spell the last word "grille"?
If not, what does the phrase mean?
"Grill" refers to jewelry that some people wear on their teeth. It is prevalent in hip-hop culture, where it is a way to show off one's "bling" - or wealth - when you smile. Your synonymous phrase is correct.
Both spellings of the word are related etymologically, as "grill" is merely a shortened form of "grille" (according to etymonline)
Green's Dictionary of Slang has "all up in someone's grill" as "aggressively confronting somebody", and "bust someone in the grill" to hit them in the mouth. It doesn't suggest any automotive origin, as "grill" is the teeth.
This is too informal for me to find in any reputable source, but in street slang around the Southeast U.S. (at least there, I can't speak for other regions), grill is used to refer to the teeth and mouth in general. This usage expanded from the concept of calling gold fronts "grills".
As an example I've heard it said, "Oh, she's cute, but she has a busted grill," which means that she has broken or crooked teeth, not damaged gold fronts.