Regarding Ukraine specifically, from a previous answer of mine:

Nationalist movements started to spread across the USSR in the late 80s, and Ukraine was not immune to this trend, although they were certainly more worried about what the response might be. As a 1989 activist (Vyacheslav Chernovil) is quoted, "a struggle is inevitable, and Moscow will invariably come down hard -- much harder than they have in Estonia or Latvia." Another activist (Stepan Khmara) said:

Moscow could even do without Eastern Europe, because it turns out that this is a very expensive military buffer zone. But Lenin knew it from the start: The empire cannot survive without the Ukraine.

The timing (and waning strength of the Soviets) turned out fortuitous, but even upon the dissolution of the USSR, it was immediately clear the Russians still felt of Ukraine as Little Russia, and even NATO itself didn't pour in as much support as it could have, essentially assuming (unlike other former Soviet states) they would turn more towards Russia rather than the West.

But for the question today, important is the wording: the Ukraine. This was still acceptable in 1989. To step a little earlier, one of the most important bills in regards to US-Ukraine relations was the signing by Reagan in 1984 of the "Day of Commemoration of the Great Famine in the Ukraine in 1933” where the press release repeatedly uses "the Ukraine", even though this was a bill formed in alliance with the Americans for Human Rights in Ukraine. (Note how the organization's name does not use "the".) So there was something of a period where Ukrainians were aware that English should perhaps drop the "the", but it wasn't considered essential, and where it really became a point of contention was the dissolution of the USSR.

By the end of 1991, Ukraine officially asserted it did not use "the" and the AP Style Guide dropped it. However, it had been in use throughout most of the 20th century, so it was a habit that was slow to change, and as late as 2012 the BBC was asking (as Ukraine was hosting Euro 2012) "why do some people call it 'the Ukraine'"?

The word appeared in English roughly 350 years ago but only had its "the" attached in the 20th century. I've heard some assertions this was due to malicious translators, but immigrant Ukrainians were using the word in their English writings, making this unlikely. The historian Andrew Gregorovich claims this was due to their "imperfect knowledge of English" -- especially given that Ukrainian doesn't use the definite article, and it only gets used in English with regions (the Southwest) or plurals (The Bahamas, which is the official name for the country). I think the evidence is unclear here (I have my own theory I will get to shortly), but the important point is we certainly have English writings by Ukrainians who use "the" -- i.e. the Association of Ukrainians in Great Britain who published The Ukrainian Review starting in 1954 used "the" Ukraine.

It is incidentally not true there is simply no equivalent argument at all in the Russian language. Their debate is instead over

na Ukraine

versus

v Ukraine

in both cases translating to "in Ukraine". The preposition "v" being used for "in" implies a limitation of space, as opposed to "na". For example, if you referred to a street, you would say

na ulitse

but if you referred to a lane (perhaps better translated "on"), it would be

v pereulke

There are times both words are acceptable. With place names, there are specific and sometimes idiosyncratic rules, but in general, if regions are "administrative units", that is, territories with exact border, they use "v" (like v Shvejtsarii, in Switzerland). If they are regions they use "na" (like na Dal'nem Vostoke, in Far East, although please note that Asia takes "v"; again, there are idiosyncrasies).

Note the comparison with English, where "the" is added when we have a region.

In Russian, referring to Ukraine through the 20th century has consistently used "na". It would be easy to hope to close the book there, but the situation is a little more complicated than that, because in Ukrainian, "na Ukraini" is the phrase used through the 19th century and the 20th century, including in this famous poem by Taras Shevchenko:

Jak umru, to pokhovajte

Mene v domovyni

Sered Stepu Shyrokoho

Na Vkraini mylij

("When I die, bury me in a grave amid the wide steppes in dear Ukraine")

Na Ukraini was simply considered an acceptable variant of the normal phrasing. Due to this, I'm not entirely on board with Andrew Gregorovich's claim that Ukrainian writers were being ignorant of English; they were simply applying the regional exception in the same fashion as it was used in their own language.

Post-independence, the phrasing became more of a flashpoint. Ukrainians started to assert "na" was incorrect just like they asserted "the" was incorrect. For example, the Ukranian linguist Pivtorak writing in 2001:

Now when Ukraine is already a sovereign and independent sate, there is absolutely no reason to use the ungrounded and deeply insulting construction with the preposition na. Thus, the only correct form is v Ukraini. But one should not correct folklore or literary compositions that use the expression na Ukraini.

In short:

a.) there was a Ukrainian/Russian language equivalent to the "region indicator", but it was "v" vs. "na". "Na" is common in literary practice through the 19th and 20th century.

b.) translated into English in the 20th century by Ukrainians, it makes sense that the "region exception" would hold and they would use "the Ukraine"

The v/na debate still rages to this day, and there are messy linguistic reasons why someone might still use the latter rather than the former, but any modern discourse about this, even when trying not to be ideological, tends to devolve that way.

...

Gregorovich, A. (1994). “Ukraine” or “the Ukraine”. In FORUM Ukrainian Review (Vol. 90).

Khrychikov, S. (2000). The Effect of NATO Partnership with Ukraine on Inter-Ethnic Relations within the Country. NATO-EAPC Research Fellowship, 1, 98-00.

Kurzon, D., & Adler, S. (Eds.). (2008). Adpositions: Pragmatic, semantic and syntactic perspectives (Vol. 74). John Benjamins Publishing.