Part of what made Chechnya different is that unlike, say, Estonia or Ukraine or Tajikistan, it was not a "Soviet Socialist Republic", but an Autonomous Soviet Socialist Republic in the RSFSR (Russian Soviet Federative Socialist Republic), which was itself a federation of republics provinces and other territories (it would rename itself to the Russian Federation in December 1991). While Union republics technically had a constitutional right to secession from the Union under the 1977 Soviet constitution, the 1978 RSFSR constitution did not explicitly provide this right to its constituent parts (although it confirmed the right of secession of the RSFSR from the union). To make matters more complicated, Chechnya wasn't even its own ASSR, but part of a larger Chechen-Ingush ASSR.

Events moved quickly in late 1991, as Chechen nationalists effectively seized control of the Chechen parts of the Republic in October, declared themselves the sole legitimate authority, and declared independence (aiming to be a full Union Republic), while also letting the Ingush part of the Republic break away and join Russia as a separate republic in November.

As u/kaisermatias and I discuss in this answer, with the manner in which events moved so quickly in late 1991, the USSR ended up being dissolved by its union republics, with part of the agreement being that they all recognize each other's union republic borders as the new international borders.

This put Chechnya's final status vis-a-vis Russia in an extremely murky territory, with the Chechen government pushing for total independence. Ultimately it would take two extremely bloody wars to determine the outcome in Russia's favor (more or less). It's worth noting, however, that because of these 1991 agreements, no other internationally-recognized state extended recognition in turn to Chechnya.