This is pretty hyperbolic. Somebody who was "moderate" in 1995 could be considered left-leaning or right-leaning today, depending on their specific politics, and in fact, without any specific context like "internet free speech", I would think a majority of them would be considered more left-leaning on average today. And even if that isn't true, the alt-right is a pretty specific flavor of hate-motivated shitposters, not just "people who are heavily right-leaning".
If alt-right has any meaning at all, I agree that it ought to refer to rather extreme beliefs. In practice, the left uses it as an all-purpose slur to refer to the half of the country that voted for the current President. (Hence my quotes.) These days, the only thing the word means is that the one uttering it is being an ass.
As a more specific example, though, Hillary Clinton was considered a moderate in the 90s. Back then she notably referred to Black youths as "superpredators". I think anyone doing so today would be seen as pretty far right. That's an example of the ground shifting.
The alt-right has a strong relationship to Nazi ideologies, "ironic" or not, and the parent is therefore reasonable in pointing out that it's an inappropriate comparison to 1995 moderate politics, regardless of left vs right opinions. The GP could have just said "far right". Now, if the parent was accusing a specific person of being alt-right, you could absolutely have a case against him based on how reasonably the label was being applied.
You're using the term "alt-right" like it is a monolithic thing, but it's not. The person I was replying to obviously wasn't using the term "alt-reich" as a nuanced description. It was a bad-faith smear.
Edit: policies that were once held by people like Bernie Sanders on immigration or Bill Clinton on crime are now considered "alt-right", but would have been considered well within the mainstream in 1995.