It can be either dative or accusative, depends on the verb. The verb "stehen" means to stand, which is the verb for a location. Locations take the dative.
If it was "sie geht neben ihn", it would be "she is going beside him" and therefore indicates a movement. Movements take the accusative.
I meant to say: Since I'm not an english native, I'm not entirely sure what you mean with your first sentence, if there's a hidden idiom or meaning I'm not aware of. If the meaning of both sentences is actually the same, duo just doesn't know about it and you could choose 'my answer should be accepted' next time. :)
In German, you would define 'she is' closer, e.g. sitting, standing, lying next to him, or would say 'Sie ist bei ihm' – She's with him, she's wherever he currently is.
Actually, that's not entirely true. There is a subtle difference in meaning. 'She is next to him' just means she is at this point in time next to him. 'She stays next to him' means she isn't leaving, so it implies a longer period of time than 'she is'.
If you wanted to say that, you would probably go with 'she bleibt bei ihm' (Menschen can fix the sentence structure, but I believe you would need to use bleiben if you want to fully retain the meaning).
As for 'is' vs 'stands', it's tricky. I guess stands implies standing up, whereas 'is' can be standing up, sitting down, laying down, etc. Maybe this is also the difference between 'steht neben' and 'ist bei'.
Sie bleibt bei ihm is correct. For your last bit: it is tricky, that's why I was unsure about your suggested translation, for this very reason. When she is with him, she might be sitting or standing or actually lying on the floor under his feet. When she's standing next to him, she's exactly doing that. And when she's next to him, she at least is by his side, so I would have accepted that as translation, less literally anway.