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Which plant you're talking about definitely makes a difference.

Which plant you're talking about definitely makes a difference. How careful you have to be with the roots depends on the species. Some are very sensitive and some are not.

I can't guarantee this would work for trickier plants, but I would try it if I could afford the risk. In fact, I'm going to. I've got some eggplant, ground cherry and litchi tomato plants that are in this situation. None of them seem to grow root noodles along the stem in humid conditions like tomatoes do, and I've heard some of them (litchi tomatoes and ground cherries) are difficult to transplant. After my soil amendments arrive (worm castings, mycorrhizae and more potassium sulfate), I can try it out. Rather than repotting after soaking, I want to plant them outside, since I'm low on space, soil, containers, and time. They'll probably be covered with milk jugs afterward (so hardening off shouldn't be a huge deal). I try to avoid hardening off where possible, but I'm not saying you should. Anyway, I should be able to add more to my answer in a week or two. Feel free to remind me in a comment if I forget.

I can't guarantee this would work for trickier plants, but I would try it if I could afford the risk. In fact, I'm going to. I've got some eggplant, ground cherry and litchi tomato plants that are in this situation. None of them seem to grow root nodules along the stem in humid conditions like tomatoes do, and I've heard some of them (litchi tomatoes and ground cherries) are difficult to transplant. After my soil amendments arrive (worm castings, mycorrhizae and more potassium sulfate), I can try it out. Rather than repotting after soaking, I want to plant them outside, since I'm low on space, soil, containers, and time. They'll probably be covered with milk jugs afterward (so hardening off shouldn't be a huge deal). I try to avoid hardening off where possible, but I'm not saying you should. Anyway, I should be able to add more to my answer in a week or two. Feel free to remind me in a comment if I forget.

With tomatoes, if your plants are healthy and are not tiny, you can just pull one of them up like a weed and repot it. (Tomatoes are very lenient with their roots.) Then water the old one and make sure soil is covering its roots. They should both live and be fine. They should develop new roots quickly. I've done this a fair amount. It works with tomatillos, too. It may stunt peppers for a while, but that can be overcome with the right nutrients and growing conditions. Potassium sulfate is an example helpful nutrient. Bright light (after the plant has adjusted a little, and not before) is an example growing condition.

For much trickier plants, or if you just want to know for sure that you'll save one plant, you might be better off snipping one off and treating it as a cutting. The other one should be fine, in this case. However, for the one you cut off, I would personally try to cut it off so low that you get at least a root or two. Then your odds of success with that 'cutting' will be much better.

With tomatoes, if your plants are healthy and are not tiny, you can just pull one of them up like a weed and repot it. (Tomatoes are very lenient with their roots.) Then water the old one and make sure soil is covering its roots. They should both live and be fine. They should develop new roots quickly. I've done this a fair amount. It works with tomatillos, too. It may stunt peppers for a while, but that can be overcome with the right nutrients and growing conditions. Potassium sulfate is an example helpful nutrient. Bright light (after the plant has adjusted a little, and not before) is an example helpful growing condition.

For much trickier plants, or if you just want to know for sure that you'll save one plant, you might be better off snipping one off and treating it as a cutting. The other one should be fine, in this case. However, for the one you cut off, I would personally try to cut it off so low that you get at least a root or two. Then your odds of success with that 'cutting' will be much better.

Alternately, you could just leave both plants together forever.