In wet areas (splash areas in kitchen, baths, etc) the use of caulk is required where tile meets a perpendicular surface. It can be hidden with the use of shoe molding or quarter round to keep the look "continuous".

Many tile setters will caulk the entire perimeter of a tiled floor regardless of setting. And then the homeowner decides if they want it 'hidden' by molding.

Caulk. I detest shoe mould almost as much as I detest quarter round. To be avoided if at all possible. It's just one more layer to collect dust and collects dings, dents & nicked paint.

Ideally the tile should have gone under the moulding and then gapping wouldn't be an issue. I'd say if you have noticeable gaps then you need to use quarter round or shoe, you don't want a thick area of caulk showing.

The base molding should have been pulled and tiled UNDER. Then re installed. Use a shoe. You can not caulk or grout. Too late for the right way now.

You'll need to match cabinet stain with a shoe or quarter round.....or ignore the gap.