I want to create a filter like search with dropdown functionality, this is a mock-up view:
Is this possible with OOTB?
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There is a feature within the list/library that can do this automatically. You need to go to the list settings and click on the Metadata navigation settings. Add the desired fields in the Configure Key Filters section. This will allow you to have a view very similar to that. It is a filter and not a search, but gets you what you need.
If you do not see this option, then you first need to enable the Metadata Navigation and Filtering site feature.
While the look is not exactly same as you shown above, Eric's suggested approach is the most viable.
If you are using SP13's in-place search for document libraries, look out for unexpected behavior of search function. For more information, see: http://the-sharepoint.blogspot.com/2015/12/strangeunexpectedfrustrating-in-place.html