Block copy paste, block value stay the same

[jimbiscuit]

Describe the bug
When a block is copy and paste, every field of the block is copied even the block field, and it is not set to the new block id.

Expected behavior
the block field is supposed to be the new id of the block and not the original one.

Additional context
When you copy a listing of a search block, the subrequest id is the same for the original block and the new one, so they have the same result.

[davisagli]

@ksuess fixed this recently for the listing block in #4239. Do we need a similar change for the search block?