That shouldn't be a problem unless you're using grounded shielded cable, in which case you should make sure the shield ground is only connected at one end.

Pull a copper wire from one ground to the other?

Potentially I might know a Gaia that can help.

[I am not an electrician, so this is not a recommendation and I would not be qualified to make one] With that out if the way, what I would do to test this theory is to measure the voltage between a ground pin in each house, using am extension cord or similar to bridge the distance. A decent multimeter should do the job. Note that the cord itself would cause a voltage drop, but the ethernet cable would too so I think that'd if you can't measure a difference with the extension cord (which should have less ressistance than the ethernet one) then it shouldn't matter(?)

Note that while ground is normally "the safe pin" (not really a thing I guess but..) in this case, where the suspicion is a differential, then it should be very much considered live. And that means all precautions should be taken, such as not following a random stranger's advice.

It might be worth looking into those plugin-sockets with ground fault protection in both ends (not sure if they'd do anything in this case), if the ethernet cable is shielded, maybe removing the shield in one end (such that the cable is grounded only to one house) could make things better (or maybe even worse?),

and while maybe not feasible: fiber wouldn't be affected by this as it's just light, and likewise WiFi or other wireless tech could be feasible. with clear line of sight, 5GHz can provide very good speed and latency.

Probably not, the costs were essentially just sticking ethernet ports on the walls next to the phone ports and rewiring the existing wires to those ports. And back when this was done (whenever we got DSL, around 2005 maybe?) fiber tech was probably prohibitively expensive. I haven’t looked up how much fiber modems cost but it would probably be more expensive even today.