I expect it to be determined by the tag that is put on the bike/box when it is checked in.
Amtrak has a Bring Your Bike Onboard bullet for "Train to Train" transfers but doesn't state a policy for Train to Thruway Coaches. Obviously check-in is a bad time to find out the trip won't work. If anyone is planning a trip like this and receives a positive answer from anyone at Amtrak, get it in writing.
Fun fact: Interestingly, Amtrak's bike box is 70x41x8.5 inches. Greyhound's box is (typically) 54x29x8 inches. Uh...
Anyway, it was the "Otherwise..." part of the question that most intrigued me. Will Thruway take your bike if you start 'Carry On' or 'Trainside Checked' w/o a box?
In an attempt to enlighten myself, I began planning a hypothetical Amtrak journey from San Antonio to Pueblo, CO (hypothetically to ride half the Trans-Am) - saving a day or more of vacation time by not going through Chicago.
https://www.amtrak.com/content/dam/p...m-Map-1018.pdf
Well, this immediately fouled my question, requiring a boxed bike. But carrying on with the exercise, I hit a glitch at El Paso: the train and bus stations are 1/2 mile apart, labeled "Self Transfer"...
Then there's the scheduled 11-hour Thruway/Greyhound ride to Pueblo.
Okay, head north to Oklahoma City & Newton Kansas, then west (compromising and getting off in La Junta). Hmm. This might work IF the third leg of the trip on Beeline Express bus will transfer your boxed bike. Amtrak and Beeline are both unclear about this so a telephone call would be in order.
Or I could hypothetically save two days of vacation by flying.