I may have a possible explanation for the two cups that we observe in this pericope.
Gregory Dix, an Anglican scholar (whose work The Shape of the Liturgy is considered useful though slightly oudated by many modern scholars), suggests in the aforementioned book that one possible Jewish origin of the Eucharist may lie in the chaburah meals which were meals occasionally held by groups of pious Jewish men for the purposes of fellowship and spiritual edification. These were governed by a sense of heightened formality and particular attention to the religious etiquette of Jewish meal practices. Dix claims that these rules that governed these kinds of fellowship meals are well known to us from rabbinic sources (although the sources Dix provides are all in German and so I cannot personally verify them for you nor even evaluate their legitimacy for myself).
Of particular relevancy to your question is the structure which governed these meals which I will summarise:
1. RELISHES Before the host and his guests 'reclined' they would be served 'relishes' - which is what the Rabbis called them - or what we may consider hors d'oeuvres. Each particular kind of relish was preceeded by each guest privately performing the prayerful recitation of a berakhah - a Jewish blessing/thanksgiving (Gk. eucharistia) - for him or herself. This is because, according to the first tractate of the Mishnah (M. Berakhot, 6:6.) as well as according to the Tosefta (T. Berakhot, 4:8 ) they were not yet considered as "one company." If wine was served with these relishes, it too would be received by each guest after personally and privately reciting the standard Jewish berakha over wine - "Blessed art Thou, O Lord our God, King of the universe, who brings forth the fruit of the vine." Dix seems to believes that in Luke 22:17-22 our Lord, functioning as the host of the chavurah meal, is serving wine to the Twelve but since this is not yet part of the meal proper, the wine is 'divided amongst [them]selves', each one reciting the blessing individually.
2. WASHING OF HANDS The guests washed their hands. After this point, late-comers are said to no longer be permitted to join because after the washing of hands and the 'grace before meals' (for lack of better terminology, please see below for details), the chaburah meal had properly begun and only those that had taken part in the washing of hands and the 'grace before meals' could take part. After this, all blessings were no longer private and individual but were to be done only by the head of the household, or leader of the chaburah or the host.
3. 'GRACE BEFORE MEALS' The head of the household, or leader of the chaburah, or the host, took bread and blessed God by reciting the standard Jewish berakha over bread - "Blessed art Thou, O Lord our God, King of the universe, who brings forth bread from the earth." He then broke the bread and distributed to the guests. This is clearly the part of the supper that we Christians recognise as what our Lord did in instituting the Holy Eucharist.
4. THE MEAL With each new fresh food being received only after the host or leader recited its particular and specific berakha.
5. 'GRACE AFTER MEALS' Though it may share the same name as the previous berakhot, this was called the benediction, the blessing, Heb. ha-berakha of which a possible Greek rendering would be ἡ εὐχαριστία (i efcharistia). This consisted of a long prayer, said by the head of the household, or leader of the chaburah or the host and consisting of a long series of Thanksgivings that recount Jewish salvation history and which structurally and in tone are similar to the pre-Sanctus Thanksgiving preface prayers in all the ancient Christian anaphoras, east and west. What is particularly striking is that this prayer begins with a dialogue between the host and the guests with the host exhorting them: "Let us give thanks..." and according to M. Berakhot 7:5, if more than a hundred people are present, the host would append, "... unto the Lord our God." Dix believes this to be the origin of the first line of the dialogue in all the ancient anaphoral Prefaces.
Of relevancy to your question is that on special occasions - perhaps on a chaburah taking place during the days of Passover or on the days preceeding it - this Thanksgiving prayer was recited by the host or leader or head over a glass of wine which was then sipped by the host first and then passed around as a common cup for all the other guests to sip from. This was termed in the Jewish literature as well as in our Pauline epistles, 'the cup of blessing.' This is clearly that "second cup" of which you and your Father were speaking and which we Christians consider part of the Holy Eucharist.
(For some interesting reading but slightly unrelated to your question, the Wikipedia article entitled Birkat Hamazon is particularly enlightening and has some extra information on this Jewish 'grace after meals', particularly the sections entitled Zimmuh and Mayim Acharonim. The section entitled Traditions is also interesting given the eschatology expressed therein as well as the "special feast" known as Seudat Chiyat Hamatim which is reminiscent of our own Eucharistic celebrations, which, it need not be said, are eschatological in nature.
6. HYMN A psalm would be sung. Matthew 26:30 and Mark 14:26 may be referring to this.
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Although I do find a lot of this to be compelling, it does still seem to me, personally, to be conjecture and good guesswork on the part of Gregory Dix. BUT, if the Eucharist, even if only in part, can be traceable to the ritual meals of various Jewish chaburot (a kind of spiritual fellowship-group) in existence in the 1st century - of which our Lord and His Twelve may have been one instantiation - then perhaps your father may have inherited a memory, passed down only orally or non-officially, that there were, indeed two 'drinkings' of wine at the Last Supper, one of which was private, represented by the private recitation of the obligatory Jewish berakha, the other of which was corporate, represented by our Lord reciting The Berakha for all. While the first was obligatory on Jews any time they intended to drink wine, whether they were alone or in the company of other Jewish men, the second was occasional and only performed in the presence of other people in the context of a ritual meal. (Indeed, Gregory Dix suggests that perhaps our Lord purposefully referred to the Cup of Blessing when speaking of His Blood so that His followers, when fulfilling His commandment to "do this," would do it only as a group, thus preserving the corporate character of the Christian eucharist from becoming, potentially, a private practice with each Christian 'eucharisting' over their personal cups of wine each and every time they wished to drink wine. )
So, given that, even in our modern day, the Jews still practice the former while we Christians continue to corporately enact the latter, it is possible that your father is the recipient of an unofficial memory of this two-fold wine blessing-drinking: a Jewish-y one which is associated, perhaps less with the OT and more with the Talmudic or Mishnaic tradition, and the other Christian-y one being inherited by the NT tradition.
Some food for thought. It is, at the very least, an entertaining idea.