1/3 The short answer is yes, but of course the real reasons are a lot more complex, and both Abkhazia and South Ossetia had very different reasons for wanting to leave Georgia.

I'll start with Abkhazia: the Abkhaz as an ethnic group are closely related to the peoples of the northwest Caucasus like the Circassians. They have been around for millennia (some Georgian historians dispute that, but they are very much in the minority), and speak a distinct language (again, related to the northwest Caucasus). Abkhazia was one of the last parts of the Caucasus to be fully incorporated into the Russian Empire, only really doing so in the 1860s, though they have long had a close relationship with Georgia.

In the late 19th century the Circassians revolted against the Russian Empire and were subsequently ethnically cleansed (deported, killed, etc; arguably a genocide). The Abkhaz were included in this, and thousands fled to the Ottoman Empire, leaving Abkhazia itself with few ethnic Abkhaz. In their place a mass resettlement program was implemented, with many ethnic Georgians moving in (mainly Mingrelians, from the region directly bordering Abkhazia). This skewed the ethnic composition of Abkhazia, and by the formation of the USSR in 1922 less than 30% of the population was Abkhaz. The rest was a mix of Georgians (35%; Armenians, 13%; Greeks, 7%; Russians, 6%, and others).

Despite the lack of an Abkhaz majority, the Bolsheviks created the Socialist Republic of Abkhazia in 1921. This was arguably done as a favour, as some Abkhaz had worked to help establish the Bolsheviks in the region. This state was an unusual creation at the time: not a full union republic like Ukraine or something, but not an autonomous republic, either; instead it was a "treaty republic", nominally united with Georgia, though that status was never clarified before Abkhazia was demoted in 1931 to simply be an autonomous republic in Georgia.

For the next five decades Abkhazia was part of Georgia, though the Abkhaz were not happy with that. Roughly every decade (1931, 1957, 1967, 1978, and 1989) there were some form of protest regrading Abkhaz rights, with the latter two (1978 and 1989) seeing the most results. Abkhaz argued that they were discriminated against in Georgia, and that ethnic Georgians were favoured for anything of relevance.

This leads into 1991 and the dissolution of the USSR. Georgia was one of the early states looking to leave the Soviet Union, and had been working on that since 1989. By 1991 they had declared independence, though Abkhazia was not interested in being subservient to Georgia. Instead they argued to remain within the USSR, and wanted to go back to the status they had in the 1920s, when they were a "treaty republic". This was not acceptable to Georgia, and in 1992 they led an invasion of Abkhazia that erupted into a full-scale war, ending in August 1993 with Georgia's withdrawal, and the de facto independence of Abkhazia, though it was not recognized by any other state, and indeed was officially blockaded by Russia. This status quo lasted until the 2008 Russo-Georgian War, and in the aftermath Russia recognized Abkhazia, and established official relations with them.

So in short, Abkhazia has had a long history of repression, and when they had a chance to get out of that, they tried to. Georgia disagreed, but was not able to re-assert their control over the region, and today Abkhazia is de facto "independent", though arguably a Russian protectorate (though Abkhazia likes to think they are more free than that).