I have used Visio at work several times in the past and have never found a feature like this. The best I've seen is when I first open Visio and am presented with all the different default stencils and I can browse through them, albeit this involves a lot of clicking.
Another option is to open a drawing and use the "search shapes" search box above the shapes on the left hand side. I found that search function useful when I had an idea of what I wanted to show, for instance, I type in "computer" and get back a bunch of little computer icons and such from across all installed stencils. It seems MS has done an ok job of tagging and naming all the shapes so they searches can be useful, assuming you have a general idea of the category of shapes you are looking for.