ID my tubeless tape

Giant?

The brown/copper colour sounds like it's probably some sort of kapton tape but that may not help you narrow down exactly what you have too much. 

From memory the DT Swiss tape I've used in the past was Kapton with a branding print on the top of it. Seemed ok in use, though would have been better without the printing on it as that just came off once exposed to sealant and probably reduced it's ability to stick to its self properly.

Peaty's tape makes a big thing out of being sticky, stretchy and slightly transparent, but I think the tape itself is branded so you would probably know if it's Peaty's 

As shoko suggests, Giant rim tape is brown/copper coloured (and works very well).

https://www.giant-bicycles.com/gb/giant-tubeless-rim-tape

As posted above, what you're looking for is Kapton tape, available in various widths from lots of places including amazon and aliexpress. 

Sounds like Giant's own tape. It's a little thicker than other tape and usually comes on a roll for each rim rather than a big roll for several rims. 

This is very timely actually as I'm having a mare taping a new rim, Stans just seems to crinkly, not sticky enough and I can't get it to conform to the rim bed. Constant slow leaks and topping up required. 

Is the Giant stuff stickier and/or more flexible, as OP suggests?

It wasn't giant tape, I thought it was caffelatex, but I now think it was spank fratelli tape, which looks to be Kapton.

cheers all