From personal experience
We have two cars in our family, my aged V70 and a fairly new Kia CEE'D for my wife. I don't have a problem with driving an older car, which may need a little consideration in the way I treat it and does have the risk of things going wrong. My wife has no empathy with a car and is paranoid about it breaking down and leaving her stranded somewhere so she insists on a newer car. (Oddly in the past when her newer car started playing up my old heap became very popular
)
When it comes to finance, her Kia cost us £8000 and I expect by the time we get rid of it in 6 years it will be worth £1000 if we are lucky, so it will lose about £1000 a year in depreciation and that's without the cost of any maintenance like brake pads and discs, exhausts, lambda sensors etc. My V70 cost me £800 about 5 years ago, Ive spent a few hundred on parts (often from breakers but new where needed such as brake parts) so that works out at about £200 per year. Comparing the two my V70 starts to look quite cheap, so should it need a few hundred spent on it to keep it going I think I might be justified doing it.
Having said that one has to be realistic about the amount one might spend, it's not a particularly rare or special model so its only value is as transport, part of the economy is that I will work on it myself (if I really mess it up it's an £800 mess not an £8000 one), should something major occur such as a total engine failure I would probably choose to have a look and see if I could find another one which looked good at a good price.
I think a lot of it is a certain mentality that refuses to be part of the discard and replace culture, and as John points out an old car is spreading the environmental cost of building it over more years so it becomes greener the older it gets.