App for collecting (and categorizing) stuff from the web
I have been looking for something similar for a while now, though I am more interested in saving bits of text. So far, I've not found anything ideal, but there are two that I've come across that work pretty well. 

Pearltrees seems to be the best, particularly for pictures and entire websites. There is an ios app for it, and you can add a button for it on your share menu. Additionally, Pearltrees is online, and both Firefox and Chrome (and probably Safari) have an extension for it. Your Pearltrees account has folders, so you can sort what you have-- there is also a "dropshelf" where you can put things until you have time to sort them later. The downside is that all of your folders and items are public with the free account-- should you want to make any of them private, you need to upgrade to a paid version.

Diigo seems to be a good way to save webpages and clips of text from a web browser. I have not explored it in detail, so I cannot speak as to its ability to save pictures. The filing system involves tags instead of files. I find this more difficult in some respects, but it may be to your liking. It also has a full version that requires payment, though you can do a lot with the free version.

If you've found anything else, I'd love to hear about it!