It's a day to day occurrence, within Web development teams who have different preferences in browser. I've witnessed it in multi companies over the last decade. If you've not noticed it, then it's highly likely you're not testing outside of a single browser. The fact that you think this is a 90s issue shows that you're either extremely junior and arrogant, or totally out of touch with modern Web development. Either that or in denial. Sorry if that sounds harsh, but if nobody says it too you then you're going to continue on in a sorry little bubble of ignorance.
If you're talking of the need to use polyfills, and test in multiple versions of multiple browsers before pushing to production, I think it's incomparable to having to write multiple native apps.
If you're talking about fonts rendering differently, or some line being some pixels further to the right, same thing; incomparable.
If you're talking about corporate web apps written for IE, those aren't accessible from a phone anyway, so the distinction between native and web app is meaningless for them.
If, instead, you're talking of websites that really don't work on a modern browser, and you stumble on them day to day, you just have an experience that's different from most other people. The easiest way to tell that what I'm telling you isn't just my individual experience is to compare what you see and read today from what used to be the case in the days of IE domination.
Acting like a dick that thinks people that disagree with you are in a sorry little bubble of ignorance is just a character flaw; nothing to do with this.
> I think it's incomparable to having to write multiple native apps.
It's not comparable to writing multiple native apps, but it's the exact same model as having to use cross-platform toolkits like QT. The web has just replaced OS with browser.
> taking about fonts rendering differently
This mimics what happens when using something like QT or Java since they at least try look kind of like the platform they're running on.
> talking about corporate web apps written for IE
IE & Chrome specific features are exactly kinds of things that make cross-browser development just like cross-platform development. Your site either has to use the lowest common denominator or be littered with platform ... err browser specific code -- exactly like native apps.