I expect it to be determined by the tag that is put on the bike/box when it is checked in. For example with some train-to-train connections I have done...

1) I traveled from Portland to San Diego using two trains; Coast Starlight and Pacific Surfliner. My bike was in a box (Coast Starlight didn't have carry-on/trainside service). The bike was tagged "SAN" so in LAX Amtrak took responsibility of transferring my bike to the second train.

2) I traveled from Austin to Washington DC using two two trains: Texas Eagle and Capital Limited. I used the trainside service. The bike was tagged "CHI". This meant, I picked up my bike in Chicago and checked it back in for my next train to get a new "WAS" tag.

I had to transfer my bike, but also had a five hour layover and went for a bike ride. If I had wanted to I could have boxed the bike and gotten a "WAS" tag when I checked into at Austin.

I expect train-to-bus and bus-to-train connections to happen in similar fashion. If they can't transfer it for you, you'll know when you get your tag...but I would be surprised if they can't transfer a boxed bicycle (or other checked luggage) for you. The reason is Amtrak also offers a shipping service where they can ship small packages - and those aren't always attended.