Spatial overlap is irrelevant for combining effects
In the DMG section on combining game effects1:
Different game features can affect a target at the same time. But when two or more game features have the same name, only the effects of one of them—the most potent one—apply while the durations of the effects overlap. For example, if a target is ignited by a fire elemental’s Fire Form trait, the ongoing fire damage doesn’t increase if the burning target is subjected to that trait again. Game features include spells, class features, feats, racial traits, monster abilities, and magic items.
This section says nothing about whether or not 2 effects are spatially overlapped. The rule applies regardless of whether two moonbeam spells cover overlapping areas or not.
Creature size is irrelevant for combining effects
If a creature is larger than medium, its space includes multiple 5-foot squares. This creates the opportunity for a single creature to be simultaneously subjected to multiple non-overlapping areas of effect, which is a bit non-intuitive. However, nothing about the rules really changes in this situation. If a creature is subjected to two effects, then those effects combine according to the above rule, as normal, even if their areas don't overlap. In particular, if the two effects are ongoing and have the same name, the creature only suffers the effects of one of them.
A creature can only take damage from moonbeam once per turn
Instead of 2 non-overlapping moonbeam spells, let's set up a line of 10 moonbeams. Then let's cast haste on a giant eagle and send it flying through the entire line. How much damage does it take? Just 2d10 damage, halved on successful save. This is because the damage from moonbeam can only trigger once per turn (emphasis added):
When a creature enters the spell’s area for the first time on a turn or starts its turn there, it is engulfed in ghostly flames that cause searing pain, and it must make a Constitution saving throw. It takes 2d10 radiant damage on a failed save, or half as much damage on a successful one.
It doesn't matter that our eagle is entering multiple moonbeam spells. The eagle takes the damage the first time it enters any moonbeam spell on its turn, and no effects occur on any subsequent entries into any moonbeam spell for the remainder of that turn, because all of these spells' damage-dealing effects are ongoing effects from spells of the same name with overlapping durations, so they don't combine. Note that this is a logical outcome, because the exact same thing would happen if the eagle flew back and forth through a single moonbeam 10 times. Also note that the result is the same whether or not any of the 10 spells have overlapping areas.
Yes, there are weird edge cases, but that's OK
Instead of having 10 moonbeam spells going at once, let's have 10 druids each ready a moonbeam spell to release right in front of the eagle's flight path. Each one releases the spell right before the eagle reaches their targeted point, then drops their concentration as soon as the eagle takes damage from their spell, before the next druid releases their spell a split second later. Now, none of the spells' durations overlap, which means the above rule about combining effects with the same name technically no longer applies, and the eagle suffers the effects of entering a moonbeam spell 10 times in a single turn. Whoops.
Of course, it is utterly illogical and rather unsatisfying that this situation should produce a different result than an eagle flying through 10 ongoing moonbeams. But I'd argue that this isn't a problem in practice, because this illogical situation is never going to occur in actual play. It's probably never going to occur even with just 2 moonbeam spells. This would require 2 druids on the battlefield, one with an active moonbeam spell and one with a readied moonbeam spell, and then a creature would, in the span of one turn, need to run through one moonbeam spell, break the druid's concentration on that spell, trigger the other druid's readied action, and then intentionally enter the 2nd moonbeam spell's area.
Remember this important quote from the DMG's introduction:
The rules don’t account for every possible situation that might arise during a typical D&D session.
If such an exceptional case as the one described above actually arises in your game, the rules expect the DM to make an appropriate ruling. In this case, it would be reasonable to rule that the eagle only suffers the effects of the first moonbeam spell, just like in the prior example of 10 ongoing moonbeams.
Another edge case occurs when the moonbeam spells are each cast at different levels and/or by casters with different spell save DCs. Once again, the rules don't explain what to do in this edge case, and it's the DM's job to fill in the blanks. If the first moonbeam spell has a DC of 17 and deals 2d10 damage while the 2nd one has a DC of 15 and deals 3d10 damage, what happens when the eagle flies through both? I dunno. Ask your DM.
1Thanks to user Scott Dunnington for pointing out that the DMG's version of the rule for combining effects specifically calls out ongoing damage effects as affected by the rule, even though each individual instance of the damage is instantaneous.