“Standby” in the sense that airlines do it will never practically work on Amtrak. On an airline, they log every passenger passing through the gate, and have a complete manifest of all boarded passengers prior to departure, usually completing the process 10 minutes before scheduled departure time.
If there are open seats due to no-shows, they will know this before the door closes, and can process passengers on the standby list before the plane leaves the gate. Plus, the gate agent will often manually verify an empty seat before giving it away to a standby passenger.
On an Amtrak train, with few exceptions (and, those exceptions would generally only be the origination station of a given train), they won’t know about no-shows until after the train has left, as often conductors will only process passengers between stops. I suppose that could work for someone boarding downline if someone upline no-shows, creating open space. But the time between confirming the space as available and a passenger getting the notice that they can board earlier would be quite tight and possibly difficult to manage (with airlines, everything is processed at the gate, and if the first standby isn’t in the area when they are ready to go, they just give the seat to the next on the list, etc.). That’s fine when you board 30-40 minutes before scheduled departure time and are done 10-15 minutes beforehand. Not so much when your station stop is only 2 minutes long.
What Amtrak could do is allow standing-room-only riders on regional services, which could effectively serve as standby for passengers wanting to travel earlier. However, this would have to be done carefully, as otherwise peak trains would just get slammed with standees, making it difficult for anybody to do anything (conductors to do their jobs, passengers to get to the cafe or restrooms, etc.). If a train is sold out, riders would just buy a ticket for another train and travel SRO on the train they want.