Fix not being able to update entities

[emontnemery]

Description:

Note: This is a copy of #19344, but against dev branch, all credit for the fix goes to @glentakahashi

When editing an entity in the frontend dialog, pressing save causes a "save failed: Entity is already registered" error. This is because the frontend always sets name and new_entity_id in the websocket command even if they haven't been changed. This adds a check that the new_entity_id is actually different from entity_id before erroring that the new_entity_id is already registered.

Checklist:

• The code change is tested and works locally.
• Local tests pass with tox. Your PR cannot be merged unless tests pass
• There is no commented out code in this PR.