I mean your legal name that can be written on passport card or used in official documents.
I use my first name 'Joshua' on the Internet, but I never disclose my last name to anyone online.
I have a personal email address as well as several other email addresses I use for websites like Italki. I use my personal email address when I need to email friends, family, my uni etc. I do use my real photo on this website, although when I signed up I was reluctant to do even that!
I was surprised some time back when I noticed how many people on this website use their real names. I've always thought it common sense you shouldn't use your real name on the Internet. Maybe I'm just being paranoid!
Why I should do that?
I can not imagine any situation that required to give my full name.
Maybe, in the case of those people that exchanged real letters.
That would be nice :) Yes, in that situations, I can give my full name and even my address.
No. For which reason I should do it? Besides if it's really needed for any official or deep frendship reason.
I would tend not to use my full name whilst surfing the web, just because many communities do not require that. But on social websites like Facebook, I see no reason not to, when the people you are in contact with are your friends or new associates.
Plus, i doubt that there is anything much that one can do with a name, regardless of it being on your passport or not. There are billions of people and many of them are going to have the same name too. Its not that uncommon.
Unless, you are afraid of stalkers and such, then it'd be safe for you to be cautious with whatever information you provide online. It could be traced back to you somehow. In that case the more pertinent aspect would be knowing who you are communicating with before any attempts to share anything private about yourself.
It's not a problem for me to give a full-name to anybody, why not))