If Tom was a wizard, he would've been affected by the Ring, which he clearly was not.
Tolkien also says this of the Blue Wizards:
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Of these Curunír was the eldest and came first, and after him came Mithrandir and Radagast, and others of the Istari who went into the east of Middle-earth, and do not come into these tales.
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Bombadil does come into the tales, ruling him out.
He does give the Blue Wizards names - Alatar and Pallando. He calls them 'Morinehtar and Romestamo' in
HoME XII: The Peoples of Middle-earth, but Alatar and Pallando are widely accepted as their names.