It seems pretty clear that searching in the Admin Center (for example Content > Media) isn't a "find this string anywhere", but rather a "find this string as a separate word". In other words, "class" doesn't match against "new_classification_of_parts", but it DOES match "new class of parts".
I think that's a poor search implementation, and it should be matching on the search term appearing anywhere in the columns searched. Don't get me wrong, I'm glad SOME kind of search is present, but it needs a little tweak to be REALLY useful.