'Zero in on' is commonly used to express focusing in on one task, goal, detail, object, target, etc. (US). 'Zone in on,' I feel like we would say this a lot less, and honestly I don't recall hearing it a lot. In principle, it would have roughly the same meaning as 'zero in on.' 'To zero' implies setting sights on a target accurately and narrowly, whereas 'zone,' to some degree is a word that implies a particular width-by-the length area, so it would inherently be less narrow of a focus that 'zeroing.' Technically, maybe, it would imply focusing on a particular area rather than on a highly focused, singular focus, but this is very pedantic. Additionally, we commonly use 'to zone out' as an expression that means to lose attention, daydream, lose focus, 'to space out,' etc.

Yes, you've got the right idea. In the US "zero in on (something)" is a common collocation, but "zone in on" isn't.