Since, in the first person singular, I can compare my situation, tastes, etc. to someone else’s saying “Neither do I.” or “Me neither.”…
… is it possible to use all the other personal pronouns in the same way?
“Us neither.” “Him/Her neither.” “Them neither.” etc.
As mentioned above, it’s all down to usage and “us neither” is not commonly used, with people preferring the phrase “neither do we” – for that reason I would stick with saying “neither do we”.
I would say the same is true of the other personal pronouns: “neither do they”, etc.