“Both are sadistic, power-hungry, charismatic and man-eating villains from animated media who have a 1930s aesthetic (Oogie Boogie is based on Cab Calloway. Who was at the height of his popularity in the 1930s; Alastor was alive around the time period and was a radio host).”
The first bit about their personalities is fine, it’s not particularly wrong, just a bit generic. That being said, the 1930s aesthetic is a pretty big stretch on Oogies end given it’s entirely based on inspiration and very little to do with Oogies character. Nothing to do with his looks, mannerism, or presentation screams 1930s. This is comparing someone loosely based on someone from the 1930s to someone dying in the 1930s. Granted, this isn’t a super important connection that keeps the MU afloat so it’s not a big deal but still not a great start.

“Both exist in magical alternate dimensions from ours filled with creepy creatures and people (Halloween Town; Hell).”
Firstly of all, what person is in hell? There’s not a single one and I only bring this up because you actually specified. Second, comparing Halloween Town to actual hell is kinda funny. Yes, both are dimensions with monsters, but that’s where the similarities end. Halloween town is a place that’s made to uphold Halloween and keep the holiday afloat. Hell on the other hand is a barely functioning society that was made by accident and is now used to punish sinners for all of eternity. Also halloween town isn’t an alternate version of our world, nothing about it is similar aside from having a community.

“Both are animal-like creatures (Oogie Boogie is a talking sack of bugs; Alastor is a deer dude).”
Not entirely wrong. Oogie is an actual sack of bugs while Al is more demon then deer but the connection still stands, it’s just kinda boring and not entirely true.

“While they are intimidating and powerful they still have minions to help them, whether they want to serve them or not (Lock, Shock and Barrel are known as "Oogie's Boys". They're incredibly loyal to Oogie Boogie; Husk and Niffty souls are both owned by Alastor. And because of that Husk genuinely despises Alastor).”
Ummmm, what? You’re comparing kids that follow oogie boogie around and do his dirty work, to what can be described as Alastors slaves. Ignoring that Alastor has actual summons that fit Oogies boys better, this isn’t a connection that goes further then “they have underlings” and even then that’s a stretch. It’s not even like it’s that good of a contrast given that, again, it’s kids who like oogie vs actual slaves.

“Both before the events of the main story they tried obtaining high social power, with one failing while the other succeeded (Oogie Boogie tried taking over Halloween Town and mix it with his own bug themed holiday, turning it into Crawloween, but failed; Alastor when he was sent to hell started killing off the Other Overlords until he reached the top and became the strongest Overlord in Hell).
Why is this framed as a connection when it’s a contrast? One that’s fine on paper, until you get into the finer details of their worlds. Oogie wanted to rule over halloween town, which would basically give him full control over the holiday of halloween. Alastor on the other hand wanted to be the strongest OVERLORD in hell. This is important cause in the grand scheme of things, an Overlord is not super powerful in hell. It’s the strongest a sinner can get but they’re like mid tiers. Alastor also isn’t the strongest overlord, with Zestial and Carmine being the strongest with it being implied him and vox are about even. It is never stated that Alastor is the strongest overlord, but that he got into a position of power very quickly. In political terms, it’s comparing Oogie wanting to be president to Alastor wanting to be a CEO, which isn’t a bad comparison but falls apart when you look at it just a little closer.

“Both became infamous in these dimensions because of this, but whether willingly or not they started to lie low (Oogie Boogie was banished from Halloween Town and so lives in the outskirts of it; Alastor after murdering every Overlord completely disappeared from the public scene for [as of now] unknown reasons).”
Ah yes, being run out of town and being banished vs basically taking a vacation. Now I’ll be nice here, lets assume that the popular theory that Alastor was lilith's lap dog for the 7 years he was gone, seeing that’s the most popular theory as of right now. It would now be Oogie being run out vs Alastor being a slave in a different dimension. Is it better? Kinda. Is it good? No not at all. And keep in mind, this is me being nice and going with the popular theory of where Al was, meaning even if it was spot on true it would STILL need to be taken with a grain of salt because it is a theory. At best, it’s a comparison thats relies on a theory to stay afloat. At worst, a shit comparison thats barely comparable.

“Both ended up showing back up when the kind hearted but naïve ruler protagonists of these worlds needed help with their plans (Jack Skellington when he got Oogie Boogie's minions Lock, Shot and Barrel to kidnap Santa Claus; Charlie Morningstar when she needed up to get the Happy Hotel up and running).”
Oogie doesn’t even show up, like at all. Not once does Oogie ever actually directly help Jack. The very first time they meet in the movie they try to kill each other. Meanwhile Alastor is one of Charlies closet friends, does everything for her, and is acting more like the cool step dad that tries to replace her real dad. Once again, that’s barely comparable.

“[IDK if I should or shouldn't remove this connection, but] Both ended up betraying the protagonists (Lock, Shot and Barrel sending Santa Claus to Oogie Boogie's lair even though they promised to not involve Oogie Boogie in Jack's plan; Hazbin Hotel hasn't gotten there yet but Viv has said that Alastor will betray Charlie).”
I’m not gonna bring up the fact that this uses something that hasn’t happened yet, my alastor MU does the same thing and it does say that it has not happened yet. No, what I’m going to bring up is that viv never said anything about Alastor eventually betraying Charlie. Yes anyone with the slightest bit of foresight knows he 100% will betray her but viv has never confirmed it.

Ending Thoughts:
As you can tell, none of these connections work well and all of them have at least something wrong with them. They’re at best stretched and at worst flat out wrong. But Connections aren’t everything, so let’s move onto the fight potential.

Alastor has quite a bit, we haven’t seen everything he can do but we’ve seen that hes a very skilled fighter, being able to work from most ranges but generally likes to keep distance with this minions and tentacles. He also has portal creation, size changing, and fire manipulation. He has an overall fencer style of fighting, opting to bait and punish if he can’t simply just overpower who he’s fighting. Now lets look at what oogie has
Oogie has…fists. And can throw some pumpkins. He also has a shadow, that can also throw pumpkins. And ghosts that do nothing. And can grow big.

Ok so one thing I think people don’t understand is everything Oogie can do is very limited given how he’s far more of a trap character with his house having his more interesting shit. Only problem is that there is absolutely NO way the fight could realistically take place there. Alastor is stuck in hell, and has no way to get out. Yes there are ways to leave hell, Lucifer and I.M.P. have shown this but alastor has no way to do so. Oogie meanwhile can’t go to hell, and even if he does go to hell after he dies, well then he just doesn’t have his shit cause they’re in halloween town.

So this would have to take place in a random place where oogie wouldn’t get his traps, so what are we left with? A sack of bugs that can punch, grow in size, and throw pumpkins. Riving I know. And that kaiju fight everyone talks about I doubt would even happen.

Yes, they can grow in size, but Al almost never does. He did so to threaten vox, and against the loan sharks to prove a point. You want to know what he would do? Summon a fuck load of tenticals to just rip oogie apart. Which brings me to the next point.

Debate: So the numbers for alastor are very skewed but lets go ahead and use his lowest end, Town Level and Hypersonic. Keep in mind this is Alastors absolute lowest end. Oogie on the other hand is…Wall Level and Superhuman. No matter what, Alastor fucking SLAUGHTERS. Now I have heard that this includes Kingdom Hearts Oogie, so lets look at that. For this lets use Alastor's absolute highest end and say that it’s true. It’s not but lets say it is just this one time: Planet and 4000c (MFTL+). Anyway oogie is Multi and IMM in speed. No matter what, it’s a complete stomp with the only “““Debate””” being if oogie would get kingdom hearts scaling.