The term בית נאמן appears in three places in Tanakh: I Samuel (2:35), I Samuel (25:28), and I Kings (11:38). A similar term is found in II Samuel (7:16).
Based on the Bar Ilan database and the HebrewBooks database, it appears that this was only popularised as a generic wedding blessing in the latter half of the 20th century. E.g. in Mishneh Halakhot (3:136) from 1959, Yabia Omer (9:43) from 1960 and , and this wedding invitation from 1960 (See lower right on the page). There were some slightly earlier usages, however, such as by R. Akiva Yosef Schlesinger in Lev HaIvri from 1924.
While the usage isn't exactly the same as in Scripture, it is related, and thus isn't purely poetic license.
Radak I Samuel (2:35), for example, explains that the term means lasting. In context there בית נאמן means, a line of kohanim gedolim that lasted many generations.
In the context of a marriage, it would presumably mean a lasting line of progeny.
See also Rambam's Hilkhot Melakhim (7:15) who refers to a בית נכון בישראל possibly in reference to lasting progeny, as he says:
מובטח לו שלא ימצא נזק ולא תגיעהו רעה, ויבנה לו בית נכון בישראל ויזכה לו ולבניו עד עולם
Interestingly, while that exact phrase is probably borrowed from Judges (16:26,29) which refers to a stable physical structure, the idea in Rambam is borrowed from I Samuel (25:28) which indeed uses the term בית נאמן.
I suspect that the term בית נאמן בישראל, is based on Rambam's similar expression בית נכון בישראל.