The Advantage is for Higher Ground, not just Height
The advantages conferred in combat by high ground are not just awarded for being above your opponent. In order to have "higher ground," you have to have a position where your opponent must fight gravity to advance.
A flying character, regardless of whether they're two inches or two feet off the ground, does not have a high ground advantage because their opponent does not need to move uphill in order to attack into their space - unless they're directly above their target, in which case they're still not at an advantage because vertical reach is easier than horizontal reach.
The same goes for a climbing creature. Unless the climbing creature is attacking from a slope, where their opponent would have to cross uphill terrain in order to reach them, they do not receive any bonus.
As for how much height would be needed assuming it was ground, that's subject to DM's discretion. I would personally say that you need at least a foot of difference (a 20% grade over 5 feet) - a single stair-step is enough to grant the bonus, but anything less than that isn't.
Large creatures - while they don't get a Higher Ground bonus just for being tall - could use their longer legs to advance over larger gaps in elevation, requiring more elevation for someone to have the high ground on them. (a 20% grade is a 20% grade, but 5-feet of it might no longer be enough) There's noting in RAW to back that up, though, so it would really be up to the DM.