TN rests specifically on Canadian or Mexican citizenship, and another nationality neither substitutes for it nor disqualifies anyone. Which passport is used does shape the process, since Canadian citizens are generally visa exempt for TN admission while Mexican citizens generally need a visa. The complication, where there is one, is procedural: a package that names one nationality in the letter and offers another at the counter raises an identity question before anyone reaches the merits of the profession or the offer.
Pick the document and stay with it
Decide early which citizenship the application relies on, and make sure the employer letter, any filing and the passport presented all reflect that choice. Switching documents partway invites questions about identity that have nothing to do with the merits. Where someone holds both Canadian and Mexican citizenship, or has prior United States records under a third nationality, ask counsel which route fits the facts.
Two records settle the choice quickly. The citizenship certificate or the passport bio-data page fixes which nationality is being relied on; the employer's letter must then describe the applicant with that same nationality and no other. Where an earlier United States record exists under a different passport, disclose it rather than leaving it to be found, since an unmentioned prior identity is treated far more seriously than a declared one.
Settle this before the letter is signed, because a signed letter is awkward to correct afterwards.