The statement that you cannot replace the barrel on a Ruger Mark pistol, and that the barrel is the serialized part, is not accurate. However, in a practical real world sense, this statement is mostly accurate.
The serialized part on a Ruger Mark pistol is the upper receiver. This receiver is a separate part from the barrel, but they are near permanently attached in a practical sense. I am not certain how they are attached besides being tightly threaded, but most shops cannot take them apart.
Historically, the only shop I know of that actually put replacement barrels onto original Ruger Mark receivers was Clark Custom Guns, and they no longer do this work. I am not sure if anyone currently does swap Ruger barrels, and it is certainly true that most people who want a new barrel will buy a new complete serialized upper that has to be transferred through an FFL.