I'm with you. I cannot bring myself to put DD's name on anything. Sure people will know her name from us saying it... mind you she has an uncommon name and often people hear it wrong the first few times... so that is in our favour! LOL
And I agree that they are ALWAYS with you at WDW vs going to school etc. So I can totally see why people feel comfortable and power to you!
That said, it is so common for kids to get separated at Disney in a split second... and at that point I would be frantic enough without the realization her name would be there for all to see. (I think we'll be going with the dog tag idea though and I might Sharpie her skin somewhere unnoticable where she can point out.)
Sure, it is nice to have the extra attention, but it gives me the willies more than that is worth. That's just me though and I'm somewhat of a worrywart.
If you are looking at stuff that is 'personalized' but anonymous, you could have the name of the character on the shirt or the name of her fave character. Or 'Birthday Girl' as PP suggested, 'first trip,' the year etc. If you had Mom and Dad shirts, you could always go with 'daughter', 'kid number' #... etc.