I had not explored this before, but found how to get around the public website displaying your real name attached to your iCloud account. Duh, go to your account and edit your name. You can change it to whatever you want. The new "fake" name will then be displayed on new shared public website links on iCloud, and you can be somewhat anonymous. I went ahead and changed mine, now relatives will scratch their head wondering who sent them pics until I tell them it's me.
Yep that's one way. I chose to create an entirely separate account in the name "dayglored". Now I just have to designate one of my iDevices to be the one that associates with that account rather than my real personal one; then I'll have the "Photos" area to play with.
In the meantime I'm also exploring the "TinyPic" site Swordmaker mentioned that he uses.
And seeing what I can do to make an anonymous site on one of my Amazon servers. Probably have to do it with an IP address rather than a domain name, since registering a domain name publicizes my real name... unless I fork over more money to the registrar to privatize it... ah, the price of anonymity. :-)