So I brewed a Northern English Brown and overshot the OG with a 1.063.
My question is, can it still be considered a Northern English Brown or does it fall into a hybrid category or something else?
What were you shooting for? If it tastes in style, it is in style. The BJCP upper limit for this style is 1.052 but who cares? Maybe call it a brown porter. Just taste it and decide then.
Probably will do best in American Brown or Northern English Brown. Unless it is really toasty and caramel like brown porter is probably wrong, and isn't a bigger beer than Northern English Brown anyway. You can and should enter the same beer in multiple categories in most competitions.
It's hard for a judge to call a beer out of style for being 10 points over, in general. What will hurt you is if you get a bigger body or more alcohol than expected. NEB should have no alcohol aroma or warmth. If you meet that criteria, and the body isn't too big, you will probably get away with it.
You can, if you kegged, also water it down. Try adding some seltzer water in the glass and if a little bit helps, mix the same proportion into the keg (or if you just want to fix 2 bottles for competition, do it in a soda bottle purged with a carbonator cap and when it is right, slowly pour into the comp bottles and cap on foam).