Yes - he is accidentally biting the collar, because it's too big.
Peter Cratchit is one of Bob Cratchit's children, a son who is older than Tiny Tim. Bob, the father, has lent a collar (of the old, starched and removeable kind mentioned in Lambie's answer) to the boy - which is 'monstrous' because it is too big for the boy - being of a size designed to fit the father.
Peter wants to 'show off his linen' - linen being the expensive fabric that collars can be made from, in the Parks. Because it is posh and he wants to show off.
Peter Crachit plunged a fork into the saucepan of potatoes - because he's very eager to try one, I think. He is probably also - hungry - this being a very poor family, before this feast occurs.
Peter may also be testing whether the potatoes are ready, with a fork - as he is the one responsible for cooking them, as described later - but maybe, as he's hungry, and suddenly now 'in charge' of that heady thing - potatoes! - he tries one.
He is now 'wearing his father's collar' - so isn't he now somehow even 'coming of age', here? The collar being 'conferred on him in honour of the day', by his father.
As Peter turned his head to put the potato in his mouth, the collar got in the way and went in his mouth instead, it means.
With 'monstrous', it is not talking about the collar being tight or uncomfortable. He is referring to it being unwieldy and flapping about and getting in the way.
The studs are also perhaps poorly applied - as this family is unaccustomed to dressing in fine clothes. That's why it is getting in the way. They have probably put it on wonky, or missed fixing some studs. Maybe the collar just won't fit onto the boy's smaller shirt.
The collar is too big, and they haven't put it on properly - in their haste - to get to dinner!
So in this scene, we have the excitement, the transformation - of the poor unhappy hungry family - suddenly bring presented with 'plenty' and being kind of goggle-eyed with it, drunk with the excitement of having suddenly 'what they always wanted'.
In modern terms, it would be a bit like gifting a beggar with an iPhone.
The collar getting in Peter's mouth is there to demonstrate the family's shift from poverty into a more abundant world, and how they are not used to it and now have to adapt to that, and are rushing - to enter their new world.
Is Dickens even saying something here like 'you can't eat money'? The boy is trying to eat his finery, albeit accidentally. Is that what finery does to you?
I think Peter is 'not proud' - as he is still doing the household task of cooking the potatoes - although he is dressed finely as 'his collars nearly choked him' - meaning he won't take them off, even when he's hot looking after the potatoes - as he absolutely loves wearing these smart new clothes!