I'm not sure that a "hard Brexit" (the consequences of which are still just theoretical) is necessarily the best comparison to what happened with the USSR. The economic unraveling that occurred in the Soviet Union happened over a longer period than just the 1991 declarations of independence, both before and after.

The Soviet Union previous to Gorbachev's reforms was a unified economic space, albeit one that was divided up into economic subregions (many of these coincided with the borders of the Soviet Socialist Republics). It's important to remember that even when central planning was in effect, there was a very strong "unofficial" side to the economic system that made it work - a factory in a region might acquire materials completely useless to them in order to barter with another factory for needed components in order to meet production targets. There was a whole group of people known as tolkati or "pushers" who used informal connections to make sure quotas were met, or to convince planners to change quota targets to more preferable amounts.

When Gorbachev started introducing economic reforms, starting in 1987, but continuing afterwards, a lot of this system unravelled: enterprises were allowed to keep a certain amount of their production for private sale, after meeting state quotas. In theory this meant that a profit motive would begin to guide industry, but in practice it had the ironic result that industries needed to appear not profitable in order to receive state subsidies (but could at the same time be informally profitable!). As Gorbachev weakened central Communist Party structures, it meant that what institutions existed for central control over local economies weakened significantly, and local party bosses, who effectively controlled regional economies anyway, had greater unfettered control, and were able to engage in asset stripping anyway. A lot of the economic back and forth between regions was mostly in terms of barter.

On top of all of this, starting in 1990 there was a so-called "war of laws" - basically Gorbachev tried to base his power in state institutions, rather than Communist Party institutions (this is when the Communist Party's legal monopoly was ended and Gorbachev created the position of President of the USSR for himself), and the state structures in the republics and regions largely followed suit. So now you had newly empowered republican governments claiming that all resources within their borders were their sovereign property, and the republics asserted their rights to make international treaties to develop economic ties and economic resources. Which is to say that even before the events of 1991, the economic system of the USSR was increasingly breaking up, and moving to something much more decentralized and confederal.

And yes, this greatly increased economic hardship as the Soviet economy began to decrease in size starting in 1989, and commodities and consumer goods became increasingly scarce (urban regions like Moscow would bargain with agricultural regions to have enough food shipped to them on bartering terms).

After 1991, it's true that this process continued apace, but I would hesitate to describe it in terms of newly independent states erecting border controls one day. If, say, in 1994 you were transporting Georgian wine by land from Tblisi to Moscow, you would probably face as many or more internal barriers within Georgia and Russia respectively than you would between the two countries. In the case of Russia effectively each federal "subject" region had some degree of control (formally or informally) over what got taxed, and how much, in its area. Over the early 1990s there was also still a "ruble zone" between the non-Baltic SSRs, which more or less lasted until the Russian Central Bank issued new rubles to replace the Soviet ones in mid 1993, and cut off supplying the republics with new currency.

I'll need to respect the 20 year rule, so I'll just mention that even after the early 1990s, there has been a whole range of various levels of economic integration between the former SSRs, from Belarus and Russia's "Union State" since 1996 (which allows citizens of either country to live and work in the other visa-free), to the recently-created Eurasian Union, to the Commonwealth of Independent States' free trade area.

So, long and short - the former Soviet Socialist Republics didn't really have a "hard" economic break with each other. If anything it's been a long, messy break and partial re-integration over the past 30 years.

ETA: additional note - the Russian Federation is the legal successor to the USSR, so while the republics pretty much continued along with their pre-1991 treaties, Russia also assumed responsibility for Union-level economic treaties, although in reality this mostly meant assuming responsibility for paying Soviet foreign debt.