Absalom was an alliance. It's silly to claim a solo kingdom is an alliance that disbanded a number of ages ago. It's equivalent to calling Pandas Covenant since many of their leaders were once in Covenant kingdoms. Even if true, it doesn't make any sense to call a single kingdom the name of an alliance that long ago disbanded.
The only reason that the Absalom talk was started was purely for PR reasons by the "Feyr Play" alliance to try and give some credibility to BB's deal break and gb into war. It sounds more noble to "declare an alliance war against Absalom" than to "deal break and gb a solo kingdom into their 1v1 war". This is the only reason why Feyr play started calling Emeriti "Absalom".