It seems this word "private" gets bandied about as an ideological assertion - insisting that we should treat this company as if it were private, completely rejecting the fact that its current form has vanishingly few private qualities.
Sure, when you put subsidies and tax breaks into the mix the line gets a bit more blurry, but legally speaking its a well defined concept, even if not all private companies match the intuitive meaning of the phrase. And yeah, Google is a very specific case, which is why there is always some level of debate about whether it (and other similar companies) should be forced to split up, and some of the resulting parts turned into a public utility service.
Google was for sure a creation of that private sector and its virtues. And even though it had some public handouts (like university research funding and the corporate liability shield), it had many private aspects - actively involved owners, a small number of employees, who were in touch with the mission, having to compete for customers, etc. But applying that term to the modern day Google as if there is a strict dichotomy of public-private seems like whitewashing the actual reality of large corporations having much more in common with government than with small businesses.