Picture would help,.all my grafts are not cut, even at 36 inches.

Konrad has good advice. Trust me, I have messed up several grafts by messing with them too soon. I like to leave them alone at least until I'm sure the graft is secure. I am small time so I can watch my grafts closely and I usually release the tape and trim after they are about 16". This will not apply to all situations but it is a place to start.

I'd like to echo the OP's question - why can't you keep two grafts on one branch?

I get it. Don't be in a hurry and the electrical tape can be removed at about a foot and a half, leaving the paraffin tape intact. Still don't understand if two grafts so close will lead to crowding. Would prefer two branches of the grafted species in case one branch is lost.

Think I understand you better,..you have two grafts @ 180 degree and you want to get rid of one?

If so, then yeas, let the better graft stay and get rid of the other,
but l let the electrical tape stay until next spring,..otherwise you could be sorry, graft can brake off in a strong wind. You might even need to support that graft on a stick or something, I never have to do in single bark grafting.

That is why I don't like heavy stock and multiple grafts,...it takes a while to heal over, heaviest I use is about thumb size and one graft.

Here is a link that might be useful: Konrad's modified bark grafting

I'll take a stab at the unanswered question of why you cut away one branch: There are a couple reasons one is that you would normally be crowding the airspace as they develop. The other is crowding at the graft location as the two grow big. The first one might be able to be solved by tieing the two so that they grow in different directions (within reason).

The second: fruit grower do not like narrow crotches. If grown that way on the main trunk it is a weak point just waiting for the right storm to split the tree. On your graft this is still something to be thinking of. There is also the concern about rot in the tight crotch. You can manage your double and get fruit from it, but have in mind that some day/year you will need to cut one off.

The reason you do two is for back up, plus in a cleft graft you may want the second one to balance the opening.

If you are grafting to well established root stock, I see no reason to cull a graft prior to dormant season. If the rootstock was large enough to make two grafts, then it will easily support both for the first growing season. Also keep in mind that lots of things can go wrong. A raccoon climbs your tree and breaks the scion...you name it.

Eventually one will need pruned off, but depending on the size of the rootstock and situation, there may be reason to leave them both for even longer.

For example, I'm bark grafting to native American persimmon with some commercial named varieties. On larger trees, I sometimes use 4 or more scions. I will have more rootstock coming of size in future years. I use these "extra" branches as "scion farms". I'll prune them heavily each dormant season and watch the bases carefully. When they get close to impinging on the keeper scion, I'll cull them completely. I'll use a stake to keep the keeper scion growing straight up while allowing the cull scions to reach out from the tree since they will eventually be removed.

The heavy pruning of these cull branches will allow them to produce great scions for future years of grafting.