Waste pipe connector identification
That kind of T-connection low down suggests that it joins the wastes from two P-traps.

It is probably more usual for dishwashers to connect above the P-trap of an adjacent sink.


An "appliance P-trap", the tapered nozzle on the left is to connect the outlet hose from an appliance such as a dishwasher.

You can also get a "standpipe appliance trap"


I've not used this type but your appliance hose would be hooked into the top.



If you can remove one of the outlets that your T-joint combines then you might be able to plan a new layout for the waste pipes that provides more space for your appliance.

It isn't clear to me from your photo, exactly what the problem is and what exactly feed the two inputs to that T-join.

In principle you can cut through the pipes and push on a new solvent-weld (or push-fit or compression fitting) joint. But you need enough pipe remaining after the cut to fit into the coupling. The more times you cut and remake these solvent-weld joints the less and less pipe you have to work with. 

As you suggest, if there is nothing using the vertical waste pipe above the T, you could cut out the T and cut back the existing pipe far enough to fit a new section of the same type of ABS pipe with two straight couplings.