I have found no way to limit the search to a particular folder. I think that SolidWorks was actually thinking about how Google search would, i.e. find a file anywhere. However, just like you, we too cannot use this search as is. I plan to discuss this with SolidWorks PDM folks at SolidWorks World.
A google like search would be fine if it would take google like operators. Put double quotes around a string of text should find that string but I get no results at all.
I thought about rolling this out to our factory who only needs to view PDFs of our drawings but that's not going to happen with these limited search options and no ability to use the search card they now use. I like the GUI but this is a showstopper, disappointing considering they spent time to create this Web2 app to replace the old Web app.
Unfortunately, looking at the "2017 What's New" there is very little development being done to Web2. More of a throw it out there so they can say "Yeah, we a have a web client" in the sales demo.
I ran into this when setting up Web2 for VPM. My workaround was to create a variable hidden to the users that is populated by Dispatch with all variables data that I wanted to search against. Once established I pointed the Web2 search to that hidden variable. This is not ideal but certainly it got us much closer to what we want.
Hope that helps!