knowledge - What world is he talking about?

" This is not easy to figure out", said Grandpa

" In this world,

403 = 308 and is same as 83"

I couldn't figure that one obviously.

" Here is a hint. 5 1 9 2" Grandpa spoke slowly.

What is he counting??

Answer

This counting with serious prompting from OP (perhaps Grandpa himself?) involves

an electronic display

then counting the segments lit-up by the spelled numbers:

403 = FOUR HUNDRED THREE = 4,6,5,6 (21) + 5,5,5,6,6,5,6 (38) + 3,5,6,5,5 (24) = 83
308 = THREE HUNDRED EIGHT is above as EIGHT is 5,2,6,5,3 = 21
so 403 = 308 and both use 83 segments.

The Hint: "5 1 9 2" Grandpa spoke slowly.

FONT (which is 7-seg here) used as clued by the initials "Five One Nine Two"

This method was also seen by Arnaud and OP deemed needed solving. (I hadn't got this.)

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Second try: In this world, what is he counting? Well, Grandpa knows

Roman numerals, so he may be counting the number of different letters when written in this form. 403 = 308 and is same as 83
403 = CDIII → C,D,I
308 = CCCVIII → C,V,I
83 = LXXXIII → L,X,I
all using 3 different letters.

And "5 1 9 2" Grandpa spoke slowly, as he was counting the

and also 5 = V uses 1 letter, 9 = IX uses 2 different letters.

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First non-standard try

Counting all (not-distinct) letters
403 = CCCCIII (usually 400 = CD, but rarely four C's is seen)
308 = CCCVIII

& 83 = LXXXIII, all having seven letters.