I've noticed the absence of the dotted i (І/і) as well as yi (Ї/ї) in the Russian Cyrillic alphabet. Why are these two letters not shared when they're vowels that are fundamental in Ukrainian?
The Russian alphabet had both і and ї in the past, but they became obsolete.
"Изменялись также правила относительно числа точек над І: Пётр отменил было их; затем было предписано ставить по две точки над І перед гласными, и одну — перед согласными; наконец, с 1738 года точка стала везде одна. Буква І отменена реформами 1917—1918 годов"
The rules regarding the number of dots above 'i' has undergone some changes. Initially, Peter I did away with them altogether. Later, it was decreed that i in front of vowels should have two dots, and in front of consonants should have one. After 1738, there was always only one dot above 'i'. The letter 'i' was made obsolete by the reform of 1917-1918.
Russian vowels come in pairs: а/я, о/ё, у/ю, ы/и, э/е.
The first three pairs are "good" ones. The second one in each pair is a front vowel (pronounced with the tongue closer to the front of the mouth than its counterpart) and can only occur after a "soft" (palatalized) consonant, [j] being one of those consonants: мяч, люк, мёд etc.
However, in writing you can see the second ones occurring after other vowels (пою), after ь or ъ (пью, объём) or word-initially (яблоко). Every time you see the usage like this, you know there is an implicit [j] before the vowel because it has to be. No Russian word uses front vowels from the three first pairs in isolation of after other vowels, and in fact most Russians are even unable to pronounce them like this.
That's why "Dr. Oetker" and "Yksi" are rendered as Доктор Эткер and Юкси in Russian, even though the first one is not even in the correct pair and the second one has an initial [j] it lacks in orginal Finnish.
The second two pairs are more complex.
Front version of э/е can exist in isolation in Russian, and all words beginning with э in Moscow pronunciation: этот, Эльза, эра and so on start with a front vowel. Furthermore, while the complementary distribution of the "good" vowel pairs is always reflected on writing (мода / мёда etc.), it's not the case for э/е: in секс, the first c is "hard" (non-patalalized) and the vowel is back, while in кекс the first к is "soft" (palatalized) and the vowel is front. Many Russians are not sure how to pronounce words like рельс, шинель, крем and can even sway between two pronunciations.
Finally, the и/ы pair stands out the most. It is in complementary distribution as well, but unlike the other ones, it's the back version of the vowel (ы) which can never happen word-initially or after other vowels.
This is also the only one pair where the front version of the vowel happens immediately after other vowels, even when stressed, without [j] or other epentheses: доил, кроил etc.
Nevertheless, even though the last two pairs are different, the complementary distribution is still there, even though it's not reflected consistently in writing.
This complementary distribution is enough to keep balance between etymological and phonetic orthography principles and avoid marking [j] in the words which never did it (because it was never there before).
That said, while Russian does use [ji] in some words (судьи, ладьи, ручьи) and so on, it never uses it word-initially or after other vowels, and, hence, there was no real reason to introduce the third letter to the pair. When still used after vowels or word-initially (in loanwords) it can be represented with йи should the need arise (Йиржи, Байи etc.)
Apparently the phonetic situation is distinct in Ukrainian and йи is used word-initially and after the vowels often enough to justify adding a third letter to the mix.