The next sentence from the document you linked gives your answer.
The ability of NEON to specify the data width in the instruction and hence use the whole register width for useful information means keeping the natural type for the algorithm is both possible and preferable.
Note, the document is distinguishing between the natural type of the machine (32-bit) and the natural type of the algorithm (in your case uint8_t).
The document is saying that in the past you would have written your code in such a way that it used 32-bit integers so that it could use the efficient machine instructions suited for 32-bit operations.
With Neon, this is not necessary. It is more useful to use the data type you actually want to use, as Neon can efficiently operate on those data types.
It will depend on your algorithm as to the optimal choice of register width (uint8x8_t or uint8x16_t).
To give a simple example of using the Neon intrinsics to add two sets of uint8_t:
#include <arm_neon.h>
void
foo (uint8_t a, uint8_t *b, uint8_t *c)
{
uint8x16_t t1 = vld1q_u8 (a);
uint8x16_t t2 = vld1q_u8 (b);
uint8x16_t t3 = vaddq_u8 (a, b);
vst1q_u8 (c, t3);
}