彼の気持ちに寄り添いたい。
I think this sentence means "I want to get closer to his feelings", "I want to understand his feelings better", but I'm not sure.
It is an inherently vague expression (to me at least), but basically it is an expression of hospitality or willingness to be kind.
So your interpretations are not off, but it means more than just understanding. It means the speaker wants to share/understand his feeling so that it helps or consoles somebody in need.
It may be close to to be compassionate in English.