Not sure if intentional, but signs can be pushed/pulled by pistons with movableBlockEntities on. Possible bug? If intentional, is there a way to stop them from being able to be moved?

Signs are block entities, and are therefore pushed. You can stop them being pushed with immovable objects, like obsidian.

well, MBE are not really following anyone else's implementation. Try to push a doublechest in bedrock and see what happens there. Carpet restricts stuff made out of obsidian and other blocks that are explicitly non-pushable. Only exception is spawners for gameplay reasons. Pushing signs is fine.

Yeah, I understand they're block entities, I just meant if carpet is trying to emulate Bedrock's MBEs, they shouldn't be on the list. Should've made that clear, sorry.

In addition to what gnembon said, I believe the classifications of what is a block entity are different on bedrock and on java.