Issue Description:

This issue is a split off from #34000, to just work on the product categories widget within the product editor.

Currently all the categories are loaded/rendered in the categories widget within the Product Editor. This can potentially cause some performance issues for stores that have several thousand categories. Not only that, once you have over a hundred categories the current screen makes it hard to find the category you are looking for.

As mentioned here, we want to display the category widget in a similar format as the Product Tags widget, with an async typeahead search to search for the categories.
Currently we already have a very similar category search field on the product list page, see the Filter by category dropdown/typeahead.

New design:
*note that the Categories -> Add design could be ignored, as the Categories -> Search field will be easier to implement and better fits with the design (cc: @jarekmorawski )

Figma: 6cBYbZmiet1l7gzc28ce8k-fi-5301%3A253895

Acceptance criteria

• Update the Product categories widget within the product edit screen with the above design.
• Selected attributes should be shown below the search field, with a small 'x' to remove them (similar to Product tags)
• A typeahead search should be added that asynchronously searches for the categories ( see example of Filter by category )
• Add new category parent dropdown should be replaced with above typeahead search as well.
• Selected children should include the parent (ex: Shoes > Hiking )

Additional context

Originally discussed in p7bje6-4if-p2.