>It’s far better to be thought of—and to think of yourself—as a project than a company for as long as possible.
'You' are neither the company nor the project, even if you are the sole person in the company. Startups are of course typically single-project companies, but the two are still not to be conflated.
If any of these things were synonymous, there would be some serious implications. If your project fails, then you as the projectcompanyperson, are a failure and may cease to exist(instead of you still being an alive person whose project failed and whose company may or may not be on the rocks). If the company is the project, then after a pivot the company is not the same company (instead of it being still the same company with a different project). But with the real concepts, definitions and identities, these are different abstraction layers essentially, and everything is more transferable, fault tolerant, and so on.
I get that with sufficient dedication and focus these things can feel like the same. I also believe that I have been someone's project.
This being said, I agree with the general analysis of the essay, specifically with regards to throwing bureaucracy to the wind in favor of lean-and-mean experimentation (and who really wants to grow up?).
It's very easy for startup culture to fall into the trap of equating the people with the project/company they're working on, resulting in unhealthy consequences when things don't go perfectly. It's a tricky thing to come to terms with, because in the early days that overlapping identity—while never healthy—is probably critical to having the momentum to carry on. But as time passes, separating your own identity from that of your project/company is critical to avoid burnout and depression; the company will always crest and trough, but you can't be pegged to its state anymore.
These companies can be quite unhealthy even when things do go well.
Fear of (personal) failure can be a strong motivator though; "One of the most interesting things we've discovered from working on Y Combinator is that founders are more motivated by the fear of looking bad than by the hope of getting millions of dollars. So if you want to get millions of dollars, put yourself in a position where failure will be public and humiliating." 
The point of incorporating a company in the first place is to create a fictional persona that is independent from the person(s) who manage it and/or work for it. To conflate a company with its owner(s) completely undermines the purpose of a company, both legally and psychologically. If you're willing to die with your company, you might as well just register as a sole proprietor.
Equating yourself with your project with your company is one of those grossly irresponsible things that self-made people tend to boast about and romanticize, along with pulling 18-hour workdays and subsisting on ramen. It's a shame that we as a society are complicit in this kind of romanticization. We should treat them as symptoms of a problem, not glorify them as rites of passage.