I think these things happen when a bad analysis is explained in a bad way.
I doubt whether the würde + past participle + haben/sein («er würde geschlafen haben»/«sie würde gegangen sein») is anything other than a würde periphrasis of the hätte/wäre + past participle construction. I believe the hätte/wäre + past participle construction should normally be preferred («er hätte geschlafen»/«sie wäre gegangen»). The reason is that the würde periphrasis is not normally used with the auxiliary verbs haben and sein. Instead, the regular conjunctive preterite (or "II") is normally used for these verbs, that is, the forms hätte and wäre.
Calling würde + past participle + haben/sein a «Futur II Konjunktiv II» seems like a particularly bad analysis to me. Instead, I would call it a periphrastic form of the perfect of the conjunctive preterite:
- The conjunctive preterite means forms like hätte/wäre/schliefe etc. (contrasting with the conjunctive present habe/sei/schlafe). It mainly expresses the irrealis mood.
- The perfect means auxiliary (haben or sein) + past participle. It expresses that an action has completed or finished. For the perfect of the conjunctive preterite, the auxiliary takes the conjunctive preterite form, that is, hätte or wäre + past participle.
- The periphrasis of the conjunctive preterite means replacing a conjunctive preterite form by würde + infinitive.
To sum it up, there are two forms of the conjunctive preterite, the non-perfect form (without auxiliary) and the perfect form (with auxiliary). In both forms, the finite verb can be replaced by a periphrastic construction with würde:
||Periphrasis with würde
||er schliefe/sie ginge
||er würde schlafen/sie würde gehen
||er hätte geschlafen/sie wäre gegangen
||er würde geschlafen haben/sie würde gegangen sein
None of these forms imply any grammatical tense. They can all be used when speaking about past, present, or future events (though of course, the perfect aspect means completion at any of those time planes).
I can only speculate why anybody would speak of «Futur» when it comes to the würde periphrasis of the conjunctive preterite. I guess it is because the verb commonly used in the periphrasis, würde, is a form of the verb werden, which is used in the construction werde + infinitive, traditionally known as «Futur». The use of the label «Futur» for that form is, I believe, a remnant from the time when every grammar of every language was described in the categories of the Latin grammar. The German werde + infinitive construction is, however, not a future tense. When speaking about future events, its use is not required. Its use does not mean that an event takes place in the future. The meaning of werde + infinitive is better understood as a modal construction expressing that at the moment of speaking, the factuality of the event cannot (yet) be ascertained.