No one can join two different Dragonmarked Houses.
All Dragonmarked Houses have signed a treaty in which they pledged their neutrality among the nations, and their independence from one another. By law, the Dragonmarked Houses must maintain their separation from one another, and that does not apply only to the dragonmarked themselves—it applies to the entirety of their operations.
Two guilds of a House would be unusual and probably not often allowed.
The organization of the various guilds themselves as described in Eberron books strongly implies that the Houses choose to organize their hierarchies in this fashion, and that there is no overlap between them. After all, generally speaking, most full-time jobs are exclusive—my employment agreement, for example, includes rules about working outside the company (and it mostly amounts to “don’t,” at least if we’re talking about a whole other job).
But that only goes for actual members. It’s plausible for Dragonmarked Houses to, at times, hire outside contractors—and those would not be required to do anything more than the job they would be hired to do. They also wouldn’t enjoy the benefits of being an actual member of a House, though. Also, it’s possible for someone to hire multiple Houses for the same overall project, which means it’s definitely possible for someone from one guild to work alongside members of another, even regularly.
Non-House “guilds” are far less rigid
The Clifftop Adventurers’ Guild is described in Eberron: Rising from the Last War as
a social club for adventurers and a one-stop marketplace for anyone seeking capable champions.
This isn’t a formal organization the way a Dragonmarked House’s Guild is, and “guild” is a strong overstatement of what it’s all about. It’s a club. The Blademarks aren’t going to really care about your membership in the Clifftop Adventurers’ Guild, and the Clifftop certainly won’t have a problem with a Blademark member.
The big thing is that the Blademarks are going to see you as theirs first. Saying “I can’t, I took an adventuring job through Clifftop” isn’t likely to fly very well if they want you to do something. Likewise, someone who hired you through Clifftop isn’t going to appreciate you being unavailable because the Blademarks have a prior claim, and Clifftop isn’t going to appreciate your sullying their name by messing up a contract like that.
As in real life, the professional way to handle that is to clear any jobs you take from Clifftop with your superiors at the Blademarks.
If not, will the Blademarks Guild resent him if he chooses the Clifftop over them?
This doesn’t really apply with Clifftop, but for the sake of argument, in a situation where you do have to choose, it shouldn’t.
While there is certainly quite a lot of pride and rivalry between guilds, ultimately we’re talking about professionals judging the professional decision of another professional—usually, people get it. Usually, everyone understands that you have to do what’s best for yourself, and that might be signing up elsewhere. Sometimes, some people don’t get it, and take it personally—in real life or in Eberron. The rest of us usually roll our eyes at that and try to avoid those people professionally.
But it also does depend on how hard the Guild you didn’t choose was interested in you, specifically. The more special you are, the more they lavished you with special promises and attention—especially if you indulged in those and “led them on” so to speak to get more—they might get more resentful. Ultimately, that’s not any more appropriate, but the frustration of having worked hard for something only to be turned down can be a powerful force, and so people might behave in ways that aren’t appropriate as a result.
It might be useful to think of it like professional sports: if some untried kid who maybe has some talent decides to sign with the other club, no big deal. If the superstar you were feting and throwing parties for decides to sign with another club, that’s more of a blow, that might be something people take personally (though they shouldn’t).