Just because this is a norm doesn't make it healthy. When your identity is rooted in something that can easily go away, it makes for a weak foundation. Lot of examples of jobs that are relatively short-lived (athletes, military, etc.) can show how difficult transitioning is when your identity is tied to your job.
I think it's unhealthy to have your identity tied to your specific employer, job title and tasks that you do for them. You as a person should not be defined by the tickets in your current sprint.
However not having a profession or any job is much different and worse. Unless you're older and retired, or independently wealthy, then it means you're going to cut back on everything. You won't socialize with other people society deems successful, you won't date, you won't grow a family, you won't travel or do anything other than continue to exist. All parts of your situation will shrink and decline. You are running out the clock toward total destitution.
I'm making this point because it's easy for people who are having a successful career to say, oh, of course I'm not my job, while missing entirely that they see themselves as a person who has a good job and will likely get another good job if their current job ends. They don't mean that they see themselves equally as a software developer or a dishwasher or on the street and it's all the same to them, so sharing the perspective that you are not your job in a discussion about extended unemployment is maybe not very appropriate.
Whether this latter reality is healthy or not, I don't know, but people have identified themselves as successful based on their trade or other social categorization for thousands of years, so at least we can say it's not new.
Will Storr writes about this much better than I can explain it here. But his point is essentially that the healthiest approach is to have your identity tied to many disparate parts of your life so that if one falters, the way you view your status doesn't hinge on that one failure. Just like you stated that "people identified themselves as successful", is a measure of status. If your esteem/status is based on that one domain, you're putting yourself at greater risk. It doesn't matter if a buggy-whip maker was the best tradesman around, he's status is still at risk when cars become popular.
The other part is that I believe research shows it's typically unhealthy to have one's social circle centered around work because those aren't very tight bonds. Again, it's a point to spread your social circle across shared interests and values rather than a job.
In most cases switching from one tech job to another will keep you in the same middle-to-upper-middle class even if you have to take a relatively large pay cut. You can even lose your job without something else lined up and it's not a big problem as long as you're confident you'll find something similar soon.
All of that is at risk with extended unemployment or being forced to indefinitely work for lower pay in a different field. Consider all the life changes someone might have to make if they have to change from being a software developer making $150k to being a rideshare driver making $40k, after six months, a year, or five years. I don't know about buggy-whip makers but probably they'd be okay with losing their jobs if they were guaranteed equal work in the new car factories. What they really dreaded was having to work for less pay in the new factories, or becoming day laborers or similar. This is what motivated the original Luddites.
We probably do disagree on the buggy-whip/factory point. Being a craftsman carries more status than being a cog in a factory. The reason why Henry Ford made $5/day a thing was workers were leaving in droves because the work was miserable and monotonous. High pay compensated for miserable work and lower status. I've known people who go from being "somebody" in a particular field (like the military) to being a "nobody" in a different field (like a factory). Even though they got paid better in the latter, they yearned for the former because of the lost status.
What your replies seem to confirm is how much we as a society base our identities on work, sometimes to the exclusion of so much else.
Also, having a job gives you a basic social identity to use to interact with people, and positive cash flow lets you be expansive and outgoing. It's very easy to take this for granted, or assume focusing on work is ultra-materialist, if you don't have a lot of experience going without. I agree that it's healthy to not worry so much about the day-to-day minutiae of a specific job, but this is a lot different from the inevitable decline that comes with not having any job at all.