Would this type of backsplash tile not require grout?
No you don't grout these period.   I have installed similar 15-20 times.   Impossible.   Grout only effective if it covers, not if it is here and there.

So number 1 you can't grout this type of tile.   They simply don't have big enough gaps.  Even if you tried you would either miss coverage or by putting too much in you will distort the layout of some of the tiles.

Number 2... I really disagree with a lot of the comments.   A kitchen is not a shower.   Unless you are in a commercial kitchen, your personal home kitchen is not really a "wet zone".   Who cares if something spills on it?   You wipe it up.  Oh no the magic tomato soup fairies are going to get underneath the tile and destroy it.   It is absurd.   

If anything it may in fact be easier to clean this tile.   Tile does not stain or discolor from a spill easy if at all... what does stain and discolor?   Yes grout.