And when people say they are “anti-tankie”, they actually mean that they are anti doing anything about the awareness of systematic inequality that tankie indicates. By definition, someone who is against change/progress is a conservative, so when someone says they are anti-tankie, they are by definition expressing a conservative stance. That is, wanting to do something about systemic inequality is synonymous with having a progressive stance on systemic inequality.
Being a tankie, on the other hand, is not synonymous with being a comunist. Tankies are just one form of communist (militant).
Other way around: communists are just one form of tankies, the word is also used to refer to anarchists and some soc-dems.