Read Exodus 31:16-17 (V'shamru) on Sefaria for the surrounding biblical/liturgical context.
The rabbis derive the obligation to make Kiddush twice from the dual command in the Decalogue: "zachor et yom haShabbat l'kadsho" (Ex 20:8), "Remember the Sabbath day to sanctify it." The verb l'kadsho ("to sanctify it") implies an active act of sanctification. The Sages applied this once for Friday night (the start of Shabbat) and once for Saturday day (during Shabbat itself).
However, the day Kiddush is theologically subordinate, Shabbat was already sanctified the night before, so the morning is just an additional acknowledgment. This is why it's so much shorter than the Friday-night version: there's no Mekadesh haShabbat paragraph, no full Vayechulu, just the biblical preamble plus the wine blessing.
The Hebrew/Aramaic name Kiddusha Rabba means literally "the Great Kiddush". But this is the shortest of the three Shabbat Kiddushes (Friday night, Saturday morning, and the optional Mincha-time Kiddush before Seudah Shlishit). Why call the shortest "the Great"?
The Talmud (Pesachim 106a) explains: it would be inappropriate to call any of the Shabbat sanctifications "the Small" or "the Lesser". So the rabbis gave the shortest one a euphemistic name. The same kind of inverted naming is used for the central Yom Kippur prayer (called Tefilat Zakah, "the Pure Prayer") and the closing service of Yom Kippur (called Neilah, "the Closing", when in fact the gates are still open).
Like the night Kiddush, the morning Kiddush is preferably said over wine in a cup of at least 86ml. If wine is unavailable, it can be said over chamar medina, any "liquor of the country" considered important in the local culture (whisky, beer, sometimes coffee in some Yemenite traditions). If even that isn't available, Kiddush is folded into HaMotzi over the bread.
In contemporary Israeli/Sephardi homes, the morning Kiddush is often a moment for grown-up children to take turns saying it, getting comfortable with the Hebrew. The shortness (~30 seconds) makes it good practice.