Can a doorbell low voltage converter share another circuit's breaker by both wires plugging into it? Or should a single wire go into the breaker but split it inside the panel using a wire but? Or does it require its own whole breaker/circuit?
It doesn't require its own breaker because it's a low load.
A breaker terminal shouldn't have multiple wires within it. Take the doorbell out, add a pigtail, and wire nut it together.
Check the breaker markings. Some breaker terminals are actually rated to accept two copper conductors. See this answer for more information. If not, you should use a pigtail.
The neutral bar terminals are likely not rated for multiple conductors, so make sure the neutral is in its own terminal.
I disagree with connecting it direct to a bus... we don't use circuit breakers for normal conditions, we use them for problems, and even a small load can have a big problem.
Apologies if I'm stating the obvious, but don't put the transformer inside the service panel, low-voltage power in the same boxes, conduits, etc. as 120/240 is a code violation. Look at the threads on the right side of your picture. It's made to screw into a 1/2" knockout hole of any common steel electrical box at any convenient location, and sit outside of said box, with 24v connections made outside the box. As such, it can simply access hot, neutral and ground from the wires passing to or through said box, and thus be attached to whatever circuit breaker supports that circuit.
If the most convenient place is at the service panel, consider a common tactic done by many electricians: putting a 120v duplex outlet right next to the service panel. It would be easy to hang that transformer off the box which holds the outlet. (I suggest a 4" x 4" 2-gang box so you have ample room to work).