Nope. The system sees the user address as [username]@[machine].dreamhost.com - nothing related to the domains you have. This user has a mailbox, where messages are stored as files - also nothing to do with the domains you have.
In addition, DreamHost has it setup that you control a single "primary" address for each user and you can have multiple "secondary" addresses by simply adding a forward-only address that forwards to the primary address. Of course both the primary and forward-only must be one of your domains.
To do that, just add a forward-only address email@example.com that forwards to firstname.lastname@example.org; this is not done for you automatically.
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