From what I remember you can use advertisements in different domains without problem.
What can not happen and the same person has more than one account on google adsense. At first for the survival of a faucet I see no other way to gain that is not adsense. For all other payments are lower with many that the faucet has costs of travel and payments.
hmm so will get adsense for a faucet that Op is offering? or any other faucet? IMO and I am a noob adsense is important for a faucet or what did I miss?
Not that it's the most important thing and that in the case faucet survive advertising. With google adsense and the easiest way to keep a new faucet, as you will have a number of initially low visits and other sources pay very little for only visits.