If Amtrak limit the policy to allow small animals that are kept in cages (birds, cats, mice, lizards, fish, etc.) and cause no disturbance there wouldnt be a problem.
The problem is, how do you regulate that, especially on trains that can take anywhere from 10-40 hours to get somewhere?
My airline's policy is small dogs and cats, under 20 lbs, able to fit in a soft sided pet carrier under the seat in front of you. The cost to do this is $75, and I think its going up after June 1st. The longest of our flights are about 5 hours, with maybe an hour added for any taxi delays.
If Amtrak had a similar policy, there wouldn't be an issue with someone taking a pet a few stops on the Regional, or some other corridor trip. But what if I want to take a pet on a trip from NYP to LAX in coach? A lot of pets can't be caged that long (obviously the ones listed above can). But what happens if one's lizard or bird or mouse gets out? This happens more then you'd think on airlines, even those like mine that don't allow such pets. I've heard many stories from flight attendants where a mouse or a gerbil or something of that nature has gotten loose and started running around the cabin, causing quite a disturbance among passengers. I can just forsee an issue where a pet gets loose, the train doors open, and it escapes, and the customer blames Amtrak.
Personally, I think a limited number of pets should be allowed, but only if one is traveling in a sleeper. It might sell more sleepers, and provided someone keeps their sleep door closed, there would be less of an issue of something getting loose. But then you'd have a roomette or bedroom that smelled like dog or cat or something that someone would complain about. Perhaps there could be a "pet roomette" or something of that nature. I'm not sure, I'd have to give some more thought to the idea.