There's a Ford Motors, Ford Model Agency, Ford Drums, Ford Gum, Ford Yacht Club, Ford Fusion, Ford Meter Box --- etc., etc. All of these different uses by different companies of "Ford" are lawful. There are innumerable other examples of multiple companies lawfully using the same trademark to brand different products, services, and companies.
Whatever word or phrase NONAME is, it can lawfully be used by lots of folks to brand lots of different products or services [as a trademark] and to brand different companies [as a trade name].
Your decision to re-brand your product [which disrupted your existing consumer's expectations and associations with your product -- and which must have cost you quite a bit] w/o speaking to a trademark attorney first was not a wise move.
To add to my colleagues response. It is also possible that NONAME Inc. was given a license by Company XYZ to use the trademark NONAME. If your product's re-branding is going well without the use of the mark NONAME, I would say stick with that and to seek the advice of a trademark attorney to register the new mark as your own.