Do websites need to look the same in every browser?

I agree, but it's so much of an oversimplification as to be almost useless.

Do websites need to look the same in every browser? Generally, no. Do websites need to tend towards looking the same in every browser? Generally, yes. Do websites need to be flexible? Generally, yes. Should we accept every browser's defaults? No. And so on.

I consider there to be a range of acceptable cross-browser degradations for a given design. That range is pragmatic; it depends on the usual variable factors like purpose, audience, budget, etc. Sometimes it's very strict, but usually tolerant to some extent.

And there are things that ought not always look the same. You want users to scale text and swap colours if needed.

In other words, the literal appearance of a website is not the only metric of similarity. Both form and function are more than skin deep. Your intent behind a design element can be realised correctly even when it renders differently across browsers.

Quantifying that kind of success is much harder than quantifying physical appearance. If browsers render the site in the same way, and the original design is known to be successful, then all the browsers will display a successful design. Maybe that's why sometimes people emphasise that approach rather than being adaptive. It's not wholly without merit, and certainly it makes sense for browsers to converge on defaults.

However, while that approach gives a decent approximation for a lot of cases (ending up within that acceptable range), I think a good designer would consider their designs in a deeper way. That is, they will judge it by how successful or effective it is, perhaps tolerating a greater degree of variation.

Even so, I think you'll find in most cases that there is only a relatively small amount of wiggle room, all things considered.

To answer with a question: How can a web page look the same on (a) a 20 inch high resolution monitor; (b) an ancient 14 inch CRT at 640 pixels wide; (c) a text browser; (d) a screen reader; (e) an iPhone; (f) a Wii; (g) an xBox; (h) a Blackberry ...

(I could go on.)

Some differences are OK, like a slight font or a tiny size difference. But I think it's best to try to perfect your site to be suited on all browsers.