Does anyone ever use the marking feature in the PC to help label certain Pokémon, like favorites, events, free to trade, good results of breeding, or anything else?

I personally never used it, but I'm considering using it like such:
Circle = Any Pokémon I personally trained
Triangle = Any favorite trades
Heart = Any Pokémon I really like
Star = Event Pokémon or legitimate legends

Any of these cases would imply I should not trade or otherwise get rid of the Pokémon.

GC: Well, when I as six, my brother told me that the heart mark would increase friendship of Pokemon. Although I know that isn't real, and I don't really find the Marking feature very useful, I do think it would be neat if markings rose friendship, happiness, etc.
GC: But hey, that's me

Cat, the markings have been around since like Generation IV or III.

I've never tried to mark my own Pokemon. The only time I really see a Marked Pokemon is when I get one off of the GTS (usually from Japan too). I guess it can be useful for organization, but I think I'll keep the cluttered mess in my PC.

Star for shinies, Diamond for event Pokémon and Circle for legendaries.

Going off the TCG's rarity symbols.
Shinies get the Star because they're the rarest, events get the Diamond because while they're not that rare, they're still rarer than the rest, and regular legendaries get the Circle because they're the most common of the three.

Since I'm going to rearrange my entire PC after I get this one last Pokémon, I'm gonna do it like- (Note: I have no idea what the markings are)
One of them (The COOLEST one) will be for legendary Pokémon
Another will be for things that I received in trades
Another for things that were hatched from eggs
And if there IS another, for N's Pokémon.

There are six total marks: star, circle, square, heart, diamond, and triangle.

I star the Pokémon which I'm guessing has the best IVs in general. Sometimes I use markings to set whether I'm already done with training for things... or whether to keep them or stash them.

I use them for my main team, and when I was breeding a load of Houndour once, I SOMEHOW managed to get two of them consecutively with the exact same everything. So they're squared. Get it?

Sometimes I use the markings to keep track of certain pokes when I'm IV breeding, but other than that, the feature goes pretty much unused by me.