The registers were 32 bit, but the data bus on the OG 68000 was 16 bit. I tend to count the latter width (I don't see the Z80 as a 16 bit processor either).

To make things even more confusing, the Z80's ALU operates on 4 bits at a time while the 68000 has 3 16 bit ALUs and so can crunch 48 bits per clock.

Half of the early ads for the 68000 called it a 16 bit processor and the rest 32 bits. The latter became more popular as time went on.