They knew they weren't good at figuring out what balance of attributes complex things needed so they copied. It wasn't their first rodeo when it came to copying something "with design allowances for local reality". Way easier to copy and alter a Fiat than trust a committee to plan a car that is a hit.
I'm not well versed on the Fiat they copied but knowing how these things usually go I can see where they'd have made changes to fit their situation (e.g. revising trivial things to use off she shelf parts already known to be available in high volume, like the soviet equivalent of spec'ing a roller blade bearing instead an odd bearing).
There are a lot of cars of that era that are like the Lada in terms of features (or lack thereof), specs, ownership experience, etc. I don't think it's anything particularly specific about the Lada that makes people like it though certainly there's some things about it that makes people with a certain worldview want to like it more than they would otherwise. I think people are nostalgic for it because it was so prolific and well cared for (or at least they were maintained) because cars weren't treated as disposable by the people who had Ladas so that makes them easy to come by and keep using today. Basically it was a decent car that got lucky in terms of when and where it existed and who it was sold to that run into a lot of staying power.