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For now, you should use the "Refine" menu or initiate the add-on filter from the "Search phrases" page.
The phrase list filter works differently in that it does an AJAX look up. However, it doesn't look at the add-on label, it will only be looking at the phrase title, on the server side. The fact that it works on the template list is sort of an added bonus in that the rows are already in a DOM and it searches all text within each row.
There's clearly an inconsistency here which isn't ideal. Not sure how that will be resolved yet, but we'll need to look into it.
FWIW, we have a dedicated menu to filter by add-on in the next release:
I'm going to call this fixed now, as these areas are consistent. The add-on ID searchability isn't really totally expected, though it does come with the standard filtering code (which will search most text in a row). The template list now uses pagination and ajax searching so it will behave similarly to phrases.
(There are also some other changes that improve how we do redirections after editing a template which should mean you get returned to the previous page, including template search results.)