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.