There's nothing in the Oath about this, which means it is up to the Peripheral Code or other Grand Tribunal rulings.
Traditionally, the charter of a covenant includes some kind of loyalty clause, so that once you sign it and become a member, you can't really join a second covenant because you can't serve two masters. But, in theory, there's nothing that would prevent a Regional Tribunal from explicitly allowing magi to be members in more than one covenant, especially if it served some larger purpose.
See, for example, the Transylvania tribunal, where the nature of covenants is wildly different from the rest of the order and described by Transylvania's peripheral code.
In other words, if you want magi to be able to join more than one covenant, that could be a clause in your local peripheral code.
I think 4e had such a ruling, perhaps in the Wizard's Grimoire? However, it might have been the bit about you can only change your lab so many times in a year and had to declare it with proper signage. I might be mixing those up.
IMHO, basically a covenant is nothing more than a contract. If you ask "could a magus contract with several societies ?" the answer is in principle yes.
Some (or many) covenant charters have exclusivity provisions in their charter, and perhaps some tribunals rule that you could not, but I personnally don't see that as genrally forbidden.
Loyalty should not be mistaken for subordination or exclusivity. One could be "free but loyal", doing what pleases him, while keeping his word. For instance, in our time you could absolutely have two jobs and be equally loyal to both employers. But you could not (normally) work for both at the same time of the day or week (because it would divert your attention and you would only pretend that you work for one of them) and you could (generally) not work as an employee for competitors as it is considered disloyal (but notice that freelancers do that commonly without a second thought).
Same goes for covenants : if you are member of two, you must fulfill your duties for both, for instance take part in the council of both, take part in both Aegis ritual and possibly give service seasons in both... the problem is to know if you can cope with that, not if you may swear two oaths.
Of course, conflicting interests would raise problems...
In the specific case being discussed, it is a hierarchical issue- a single covenant is establishing "child" covenants that it does not want to be chapterhouses, but the leaders of these covenants would be members of a single covenant as well to keep the overall group united. In this way it is more like being a part of both the national party and the local party organization in politics (at least in the US, I'm not sure how well this analogy translates)
Perhaps this can be easily resolved by forming a Societas rather than a "super-covenant." Everyone is a member of their own covenant, but also a member of the over-arching Societas. This sidesteps any concern over belonging to more than one covenant, and a Societas can have any membership requirements it's members agree upon, including a loyalty oath.
As an odd case, the covenants of Transylvania are pretty strictly agreements - not locations. Members are of particular Houses and inclinations, and agree to certain (some seemly arbitrary) obligations and restrictions.
For a group of magi living together toward a communal end, they have oppidum, mostly (always?) members of multiple covenants. Oppidum are nominally temporary arrangements, even if hundreds of years old.
Here is a case of covenant members living with members of other covenants in a community, generally in such a way as to not conflict (although conflicts between covenant and oppidum requirements might arise).
I'm sure I read something somewhere (not Against the Dark - probably HoH:TL, but I can't find it now) about all Tremere being automatically members of Coeris, but being able to automatically drop that status if their covenant required single membership then take it up again the moment they left.