You can if you want, it is a choice, especially if it is a consumer of a versioned API, just like you would version a mobile client.

One counter-example is Reddit's old.reddit.com interface, compared to their new one.

I have seen some older Software/SAAS UIs displaying version numbers. But letting you switch may not make sense because usually a change happens both in backend and frontend and switching versions is not something vendors want to offer due to the complexity of switching required.

We've been versioning UIs since 2015, after dealing with a bunch of weird bug reports that we tracked down to people keeping tabs open for months. Needs careful backend planning and backwards compatible data handling, but it's a solvable technical issue. API calls from the front-end can send the active version with calls, letting the backend understand that something older is calling and do version upgrades of envelopes and payloads, and the front-end needs to be designed to ignore extra fields, but all of that is testable, and once things are in place they give you both the protection against weird inconsistency bugs, ability to experiment (running a/b versions in parallel) and detailed monitoring to troubleshoot weird issues.

It's quite sobering to look at the list of active versions and see a long tail of what's actually connecting to our backend.