Suppose that someone browses non-Youtube videos (like on Vimeo) in Google Chrome's Incognito mode.
Does Google collect and store any data about this activity ("watched videos on Vimeo" activity)?
In general, does Google store ANYTHING that is done in Incognito mode?
I think you need to distinguish between "Google" and "Chrome".
Chrome is a browser and the main feature of the Incognito mode is to delete any locally stored information from the browser session after the Incognito mode was closed. The point here is locally stored because this is all the browser can fully control.
Google is instead a company which among others things collects information about the users behavior by being included with Google Analytics or Doubleclick into many websites. This is similar how other companies like Facebook or the various ad and tracking networks are included into the websites. And this kind of data collection is also independent from the browser you use, although some browsers have special features or some extensions can be added to reduce the amount of tracking and profiling.
This remote data collection does not stop when the browser is in Incognito mode. In fact, usually these ad and tracking networks are not even aware that Incognito mode is used. What is different though is that tracking information from the "normal" mode and Incognito mode cannot be easily associated with each other, so the profiling done in Incognito mode is mostly independent from the profiling done in normal mode or from profiling done in other Incognito sessions.
Advertising vendors can and do keep and collate every bit of information they can, which is a lot, and can (and therefore presumably do) associate your activity in Incognito mode with your other activity.
This can be done through your IP address, together with machine-specific information (for example operating system and screen size), which will be the same for both.
Information they can use:
This is more than enough to distinguish between devices using the same IP address. Once they've identified the device, then can associate it back to you, and back to your other devices, via your normal browsing.
So it's possible, and because it's possible, I assume that they do it, since that's how they make their money.
And since Google is first and foremost an advertising vendor, I include Google in that.
The short answer to your question is: yes, Google does collect and store data about activity done in Incognito mode. It might not store data about all activity done in Incognito mode, but certainly some. Not because “Google Chrome”, the cross-platform web browser, is developed by Google, but because – as MooingDuck already pointed out – almost every website you visit will send data about your visit to Google, Facebook, and/or other advertising companies, mostly via third-party trackers. To what extent this takes place depends on your browsing habits, what sites you visit, if you are logged in to your Google account, your device's and browser's privacy settings, etc.
For more information you can have a look at this video titled “Understanding Digital Tracking” – but be aware, that YouTube is a company owned by Google, so Google will certainly track access to that video too.
To extend other answers and give you a little more information on how this "remote tracking" works:
as @Ben answered, when you visit a website there are plenty of information sent to this website, not only the IP, but also screen sizes and versions of software/system used. Amount of this data is exactly why an incognito mode does not provide you with any extra protection from being identified. An example you can test and verify yourself is here: https://www.nothingprivate.ml/ - enter your name (or whatever), then fire up incognito mode and check if your name from non-incognito is correct. The backend here just stores a combination of your User-Agent string, your screen size and other stuff Ben mentioned, along with your name (or whatever).
Above is one of the reasons why a Tor Browser tries to keep everyone within the same configuration set. It even displays a warning when you change default resolution of the window - because then you stand out.
One thing incognito helps in is that it does not send any cookies that are saved in your regular browser. This certainly limits the 100% confidence of identifying you, but a big problem for general tracking anyway, unless there are a lot of people with exact same setups in your IP block area.
TL;DR - using Incognito Mode offers more privacy in the google Chrome browser than not using it. The feature is intended to hide your actions from those who share your machine, but does little to hide you from the all-seeing googly-eye.
Does Google collect and store any data about this activity ("watched videos on Vimeo" activity)?
A quick look at the Vimeo homepage source shows resources requested from googletagmanager.com and googletagservices.com - so, yes, if you have javascript enabled then google are getting at least something from your visit.
In general, does Google store ANYTHING that is done in Incognito mode?
Private Browsing systems are generally built around the concept of hiding your history from your machine. Not on stopping the flow of valuable information to google. If you have 'SafeBrowsing' enabled for example then effectively the URLs you visit are sent to google even if they aren't logged in your history.
Google's Chrome Browser Policy for incognito mode is conspicuously silent on Installation Tracking, Promotion Tracking, or Field Trials, all of which send potentially personally identifiable information to them.
How Chrome handles your incognito or guest information
Cookies. Chrome won't share existing cookies with sites you visit in incognito or guest mode. Sites may deposit new cookies on your system while you are in these modes, but they'll only be stored and transmitted until you close the last incognito or guest window.
Browser configuration changes. When you make changes to your browser configuration, like bookmarking a web page or changing your settings, this information is saved. These changes are not affected by incognito or guest mode.
Permissions. Permissions you grant in incognito mode are not saved to your existing profile.
Profile information. In incognito mode, you will still have access to information from your existing profile, such as suggestions based on your browsing history and saved passwords, while you are browsing. In guest mode, you can browse without seeing information from any existing profiles.
This does not include the myriad of other google services which may give less detailed (but still very useful) information such as google DNS and DoH.
For all their talk, google are serial privacy abusers. In the last 12 months they've come under fire for attempts to track users between sites without 3rd-party cookies, collecting personal data without authorisation, failing to wipe cookies when told to, and privacy concerns in reCAPTCHA. Google's concept of privacy is that it is privacy from one another, not privacy from google. As Eric Schmidt famously said...
If you have something that you don’t want anyone to know, maybe you shouldn’t be doing it in the first place.
...and that was in 2009. What makes the google search product (and powers their ad platforms) is really the amount of data they have on internet users.
It's also worth pointing out that switching browsers is unlikely to be that helpful as many browsers use the Chromium codebase. When Microsoft re-worked Edge to use the Chromium engine they had to alter or remove over 50 services that supply data to google.
