I don't agree that the same term is used for all those things. The word "tissue" is shared between them but there is almost always another word attached to it to indicate what you're talking about.
The thing you blow your nose on is a tissue or a pocket tissue.
The stuff you use to wrap presents etc is tissue paper.
The stuff you use in the bathroom is toilet tissue, toilet paper, or toilet roll.
As Piscean indicated, the brand name which is used generically in some countries is "Kleenex". This is not used in the UK. The brand is, of course, sold here but it's never been used as a generic term for tissues. Not so long ago, at a yoga class, an American girl leaned over to me and whispered "Do you have a ...?" I had to ask her to repeat herself three times before I finally realised she was saying "Kleenex". Had she said "Do you have a tissue?", I'm pretty sure I'd have understood her the first time.
The same goes for the little sticky thing you pop on a small cut on your body. While on holiday in the States a couple of decades ago, someone asked me "Do you have a [unintelligible to me]?" Again, after three or four attempts, I finally realised she was asking me for a "Band-Aid". I had never heard the brand name used as a generic term until that day. In the UK, it's just "a plaster".
