One of the tasks of a signer is to determine whether it should sign something. There are a few different schools of thought on what should be done for this determination, however many hardware signers follow the pattern of parsing the transaction to be signed, showing it to the user, and asking the user for confirmation to sign it.
Even with many wallets doing this basic pattern, they still do some things differently, such as what information should be shown to the user, and what can be assumed about the user. Generally, the assumption is that the user is not technical and so as much technical information needs to be elided or otherwise separately verified by the signer to be safe.
For example, a technical detail that is often hidden is that transactions have change outputs. One of the most basic things for a hardware signer to do is to show the user all of the addresses that will receive Bitcoin in the transaction and the amounts that they will receive. But most users don't know about change outputs, or they don't know what change address is theirs. Showing the user the change output may confuse them, so the hardware signers will determine which outputs are change so that they do not need to show them.
But how can a hardware signer determine which outputs are change? For single sig scripts, this is easy as the hardware signer can generate the key for the script, and by assuming that anything that it can sign for is change, it can find the change outputs.
However for multisigs there will be keys that it does not know nor can generate. For multisig inputs, it needs to be told the script (and therefore the other keys). It needs the same for any outputs that are change. But even if it can produce the change output, how does it know that it actually is the change output? How can it be sure that there wasn't malware that swapped out some of the keys for an attackers?
Some hardware signers deal with these questions by requiring the user to have registered the multisig on the device before they sign any transactions involving it. The user provides the multisig threshold as well as xpubs of the involved keys, and the order they should appear. The wallet may store this or they may provide a signature of sorts that would later prove to the device that it had been pre-registered. Then during signing, it can look at that multisig information in order to determine which output is change.
Other wallets do not necessarily need this information and instead try to figure out the change from the transaction itself. A PSBT can contain enough information to figure it out. The signer can look at the inputs, determine the threshold and the xpubs of the multisig from the inputs, then check if any of the outputs match that. Thus they can determine the change output.
Change output detection is not the only problem. Some wallets want to do further verification to ensure that users are not accidentally losing funds. Suppose the user is participating in multiple multisigs. Some hardware signers want the user to make sure that the user knows which multisig they are spending from. This requires such devices to store the multisig or have be able to verify a commitment to it, and so require registering the multisig before signing.
There are also other wallets that simply don't care to verify and show this information to the user, and may just blindly sign anything that they can. It all depends on the security model that the hardware vendor is trying to go for.
Here are is a non-exhaustive list of hardware signers that require registering the multisig:
- Ledger (Bitcoin App 2.0+)
- Coldcard and clones
- Jade
- BitBox02
Some hardware signers that do not require registration:
- Ledger (Bitcoin App 1.x)
- Trezor and clones
- BitBox01
I don't know about the seed signer, but from the documentation I can find, I think it does not require registration.
It's important to know that this is all just implemented in software on the devices that can be updated. As you can see in the list, Ledger has changed from previously signing anything, to requiring registration in order to sign. Other signers may choose to implement registration as well.