To enhance the user experience with a surprising icon set, I was thinking of using 3d icons, but most icons are small, flat, and generic. I understand convention is important for recognition, but is there a reason why the trend is 2d on the web and 3d for desktop apps?
Ease of comprehension, integrity at smaller sizes
The answer may lie in this point from the NNG article on icon usability.
Keep the design simple and schematic. Reduce the amount of graphic details by focusing on the basic characteristics of the object rather than creating a highly realistic image in order to speed up recognition. (Intricate details are difficult to distinguish at smaller sizes.)
Nielson Norman Group, 07/2014 - https://www.nngroup.com/articles/icon-usability/
The more detail, the longer it takes to parse, and the more likely the essence of the graphic will get lost when displayed smaller sizes e.g. on mobile devices.
This isn't necessarily just about style, but it also ties back to the branding and image that the companies are trying to present to the users or their customers. You can certainly try to style icons so that it still has a consistent style, but because the 3d shapes and patterns creates a different look and feel compared to the minimal and simple icons in the 2d and flat style.
Most web applications want to provide a user friendly and easy-to-use interface, and therefore they try to match the icon design to the rest of the intended user experience of the application.