I like the ability to filter by tags, but not sold on the 2 search boxes. What about using the dropdown filter menu like in content for tag selection. See screenshot (Replace dropdown with your normal tag selector)
The reason is that:
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The behavior is not consistent. The content search makes a full text search by default, the asset filter a contains query.
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Whenever you select a tag you also have to replace the input value with odata query, which is hard to understand.