There's no "8-bit pixel resolution". The 8-bit means there's only 2^8 = 256 colors in single image. Depending on console, those 256 colors were either pre-selected or you could define your own colors (even modify them in real time).
Those concepts are loosely carried over to modern pixel art. In most cases 256 colors is enough for static pixel art, especially if you avoid gradients (as most artists do).
When talking about graphics that's correct, but I think in this kind of context the "8-bit" refers more to (nostalgic, rose-tinted memories of) the visual style of so-called 8-bit consoles, in particular the NES/Famicom, and most consoles that we consider 8-bit didn't allow for anywhere near 256 colours on-screen. Here the 8-bit qualifier refers to the 6502 CPU's word size; the NES in-particular only had a total palette size of 54 colours [1], and additional colour limitations meant that in practice you'd rarely see more than around ~25 at a time.
Or take the Sega Genesis/Mega Drive, a 16-bit console; its colour capabilities didn't give anything close to 2^16 = 65536 on-screen colours. Instead you got a 9-bit palette with only around 64 on-screen colours [2].
GIF only supports an 8-bit colour palette (256 colours per frame - albeit colours can be picked from a 24-bit colour space), so it's definitely an 8-bit colour palette.
There's no such thing as an 8-bit pixel resolution. In fact even amongst the 8-bit consoles, their output resolutions differed: NES: 256x240; Master System: 256x192; Game Gear & Gameboy: 160x144; and the Atari 7800 supported a few different resolutions.
