If the term is generic then Alphabet loses a right to prevent others from using it in the marketplace. "Kleenex" is not a legally genericized trademark in the US -- though it certainly has potential to become one.
Actual genericized trademarks in the US include: Aspirin, Dry Ice, Flip Phone, Kerosene, Thermos, Trampoline, Videotape. You absolutely can sell your own off-brand of the above product using these terms in the US. Entirely legit.
It all depends on what the company that owns the trademark does to try to prevent erosion and maintain a strong tie between the word and the brand itself. A related Wikipedia article [1] specifically mentions the lengths Google has gone to in this respect.
All that said, there may be an issue of prior art, from The Hitchhiker's Guide to the Galaxy:
> "And are you not," said Fook leaning anxiously forward, "a greater analyst than the Googleplex Star Thinker in the Seventh Galaxy of Light and Ingenuity which can calculate the trajectory of every single dust particle throughout a five-week Dangrabad Beta sand blizzard?"
Prior art applies to patents, not trademarks. Trademark rights arise from use in commerce.
Trademarks generally go to the first user within a given market. In this case, had the Googleplex delivered online search services and were in business before Google they would be able to stop Google from being "Google" if that it would dilute the value of their trademark or, more likely, cause customer confusion.
Ideally they would have federally registered the trademark and also protected it in the market in a timely fashion. If not they would lose their protection, which is why trademarks are vigorously protected.
Actual genericized trademarks in the US include: Aspirin, Dry Ice, Flip Phone, Kerosene, Thermos, Trampoline, Videotape. You absolutely can sell your own off-brand of the above product using these terms in the US. Entirely legit.
More here: https://en.wikipedia.org/wiki/List_of_generic_and_genericize...