The correct use of 'another' is singular, only
The word 'another' in English is actually a contraction of 'an other'. It is properly restricted to referring to a single alternative. In this sense it is helpful to think of it as meaning 'one other', and not meaning 'any others'.
Thus, when a swarm says:
The swarm can occupy another creature’s space...
By grammar, we know this means that the swarm can occupy the space of one other creature (and note that creature's is the singular possessive as well). If there is a space with two creatures already in it (either two swarms, or a swarm and a non-swarm creature), then the addition of one more swarm would make it the third creature, and would not be permitted under a strict definition of 'another' (or creature's). If swarms were intended to stack, their ability should be written "The swarm can occupy a space with other creatures".
But 5e rules are not always written with correct usage
However, as I am frequently reminded by other site users when I attempt to analyze the language of 5e rules text here, the designers do not always adhere to accepted prescriptive grammar and style, and it is often doubtful whether they attempted to do so. In the PHB we can find cases where "another" clearly means one, singular other, but we can also find rules where it simply does not.
Consider:
Spells Known of 1st Level and Higher
Additionally, when you gain a level in this class, you can choose one of the bard spells you know and replace it with another spell from the bard spell list, which also must be of a level for which you have spell slots.
As written, this says that you can un-know one bard spell and replace it with a single other bard spell, but not with multiple other spells. Clearly it is the intention of the designers here to conform to the grammatical rule that 'another' means a single other.
But contrast this with:
Totemic Attunement
Bear. While you’re raging, any creature within 5 feet of you that’s hostile to you has disadvantage on attack rolls against targets other than you or another character with this feature.
As written, this says that if our target was surrounded by three raging Bear totem barbarians, the target would be able to attack only the first and second barbarians regularly (you and one other), but the third one would be attacked at disadvantage. Clearly here the rules are using 'another' to mean multiple 'any others' rather than a singular 'one other'.
This is even more clear with:
Object Reading. Holding an object as you meditate, you can see visions of the object’s previous owner. After meditating for 1 minute, you learn how the owner acquired and lost the object, as well as the most recent significant event involving the object and that owner. If the object was owned by another creature in the recent past (within a number of days equal to your Wisdom score), you can spend 1 additional minute for each owner to learn the same information about that creature.
Here, despite the rule writing "owner" (and not owner(s)), and despite 'another creature' properly meaning one single other creature, the use of 'for each owner' makes it clear that in this passage 'another' means plural, 'any others'.
Thus we cannot rely on the correct grammatical usage of 'another' to tell us whether or not swarms can stack, because the PHB does not always conform to this usage. There is another grammatical clue - the use of 'vice versa':
The swarm can occupy another creature’s space and vice versa...
"Vice versa" ('with the relations changed') means that if we switch 'swarm' and 'creature', the rule still applies. If 'another' here indeed meant multiple creatures, then the regular rule means that a swarm may occupy the space of multiple creatures, and the order-changed rule means that multiple creatures may occupy the space of a swarm. That is, take two non-swarm creatures. Normally they cannot occupy the same space. However, they can both occupy a space so long as the space also has a swarm in it. Somehow the swarm permits other creatures to fit together. Since that is clearly not the intention, the use of 'another' in this case must be singular, right?
RAI
Unfortunately, no. As stated before, we cannot trust that the designers wrote the rules with correct, prescriptivist grammar in mind. What we have here is ultimately a case of RAI interpretation, not RAW. That being said, consider our alternatives. Accepting 'another' as plural in this case means that unlimited numbers of swarms will fit in the same space. Accepting 'another' as singular means that we are limited to one swarm in a space in addition to the creature already occupying it.
This DM, at least, is more comfortable with a limit of one than with no limit at all.