Also, no, the basics of supply and demand will not make cheap cars a thing. It's not the cost of a car that you can't afford, it's the cost of maintenance. Fuel, maintenance, and insurance (mandatory in the US btw) are all costs to use the car - the average being around $100/mth.
https://newsroom.aaa.com/tag/cost-to-own-a-vehicle/ -> 1.1k
And also no, new cars will not become cheaper over time in the US. There's already 19% of the US families who make under 24k a year (we roughly double UBI to account for 2 adult UBI incomes and one car per household rule). That's 24 million households already out of the 130m households in the US. The market for used cars is actually quite big (17.6 million transactions / yr) which is probably why there isn't an $8200 new US car offering (which, btw, requires savings in order to purchase).
Back to my original point, if you have a budget of 1k/month, you could spend ~$400 on rent, $300 on food, $100 on your car, $100 on personal needs, and $100 on misc / emergencies in a small suburban town (like the outskirts of Pittsburgh) -> I've done this in college before. But realistically you would need an extra ~$600-1000 a month to pay for things like - internet, health insurance, furniture, clothes, laptop, phone, heating / ac bill, eating out every so often, a social life, hobbies, travel, etc. So yes, you'd still want double of your UBI income.
The kind of people that live on UBI (with no other job) are not the kind of people that care about savings.
Second, you ignore the basics of economics of demand and supply: if a large enough proportion of the population lies on UBI you will even find cars cheap enough for them, either second hand or purpose built. True story, Renault is a car manufacturer that sells their cars for prices very similar to everyone else, but they built for some years a model that was selling for 5000 Euro (about $6000 at that time) new, with a modern and solid engine and all the safety features mandated by law in Europe. That car was also sold in France for a lot more (8000 euro or more) just because people in France afforded to pay more. Now Renault is still selling 7000 Euro cars (Dacia Sandero) that are really good for the money, while selling Renault branded cars for around 20,000 Euro and more.