It's probably a coincidence.
Preface: My main source for this entire answer is The Origin of the Italic Imperfect Subjunctive by J. H. Jasanoff (Vandenhoek and Ruprecht, 1991). If this is contradicted by more recent scholarship, please feel free to correct me!
All but one of the example words given are in the second person singular, because this shows off the vowel lengths more reliably (and because it's what Jasanoff does).
The imperfect subjunctive in Latin was formed from a Proto-Italic morpheme reconstructed as *-sē-. Where this morpheme comes from is unclear, since it doesn't line up with anything in PIE, but it doesn't seem to be related to sim, sīs, sit (Proto-Italic *si-ēm, *si-ēs, *si-ēd).
In most verbs, this rhotacized: *amā-sē-s > amārēs, *face-sē-s > facerēs, and so on. But in the rare athematic verbs, the s appears as itself or assimilates: *es-sē-s > essēs, *vel-sē-s > vellēs.
When I was first learning, I was taught to form the imperfect subjunctive by attaching personal endings to the infinitive. But this isn't correct either: the infinitive was formed with an ending *-si, with a short vowel. So using this rule will get the vowel lengths wrong.
Future Perfect Indicative
This tense comes from "the S-future of the perfect". In other words, the future morpheme -(i)s- attached to the perfect stem. (The "S-future" is very common for the actual future tense in Ancient Greek: present lyō → future lysō. But in Latin, it was replaced with a relative of fiō in the first and second conjugations, and the original subjunctive in the third and fourth.)
So a form like amāveris comes from amā-v-is-(i)s. The similarity to eris seems to be coincidence: vowel reduction and rhotacism turned -is-is into -eris. But I'm not sure where eris itself (as in "you will be") comes from, so I can't be sure they're unrelated.
This comes from the optative of the S-future of the perfect: the optative morpheme -ī- added on to the future perfect stem. The long vowel distinguished this from the future perfect indicative at first, but as Alex B points out, the distinction vanished by Classical times. So this amāveris comes from older amāverīs, from amā-v-is-ī-s. Once again, the similarity seems coincidental.
Notably, the S-future of the present used to be used for the actual future in Old Latin (faxō "I will do" < fac-s-ō is attested), while the optative of the S-future of the present was used for the present subjunctive (faxīs "you should do" < fac-s-īs).
This one is formed from the subjunctive morpheme *-sē- (as seen in the imperfect) combined with the S-future of the perfect (as seen in the future perfect). Amāvissēs comes from amā-v-is-sē-s, and the additional consonant prevents both vowel reduction and rhotacism.
So this one is actually half-cognate with essēs: the -sēs ending is the same. But once again I was taught wrong: the perfect infinitive comes from amā-v-is-si, with a short vowel. That -si is (probably) not related to the subjunctive -sē except by coincidence.
TODO: I don't actually know where this one comes from.