Things like this is always up to a GM and clearly in homebrew territory, but this could be the very definition of "Improvised Weapon".
Sometimes characters don’t have their weapons and have to Attack with whatever is at hand. An improvised weapon includes any object you can wield in one or two hands, such as broken glass, a table leg, a frying pan, a wagon wheel, or a dead Goblin.
Often, an improvised weapon is similar to an actual weapon and can be treated as such. For example, a table leg is akin to a club. At the GM’s option, a character proficient with a weapon can use a similar object as if it were that weapon and use his or her proficiency bonus.
An object that bears no resemblance to a weapon deals 1d4 damage (the GM assigns a damage type appropriate to the object). If a character uses a ranged weapon to make a melee Attack, or throws a melee weapon that does not have the thrown property, it also deals 1d4 damage. An improvised thrown weapon has a normal range of 20 feet and a long range of 60 feet.
Important questions: Does this count as a polearm weapon, a basic improvised weapon (d4 damage) or something else, like an actual bear trap?
A normal bear trap also causes 1d4 damage so it seems that dice fits the standard improvised weapon characteristic. In this case however there needs to be some ruling on how a victim of it could still break free of the player's hold without removing the trap. Just because he actually hits something with it and the trap snaps the victim could still avoid being dragged around, and instead disarm the character (Something like how a fish can move off with an angler's fishing rod)
I personally would require: 1. an attack roll as using an improvised weapon (is the character proficient?) 2. If the attack hits see with DC 13 Dexterity save for creature if it actually gets stuck into it (or just hit for simple damage with the pole), then 3. allow the victim to use its action to make a DC 13 Strength check on its turn to see if it can in a way break free (Each failed check deals 1 piercing damage to the trapped creature). At last 4. an additional Strength check should by the player be required to see if it can hold on to the pole and compare that to the first Strength check of the victim when it rolls the check on its turn. If the creature rolls low enough to remain trapped but higher than the Strength roll of the player's character then the victim creature would still be trapped in the actual trap on the pole but disarm the player's character, with the pole dangling in the air. If the creture remains a victim of the trap and firmly held it would be considered grappled.
As GM you could rule that he can use stronger, specially made traps as he improves upon this weapon/trap idea, maybe by using different materials (adamantine?) which increases saving roll DCs and initial trap (but not pole) damage. Consider handing the pole damage (not the trap) as 1d6 (between club and quarterstaff dice).