Does Google collect and store data about activity 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:

• IP address
• Operating system and version, and browser version
• Screen size
• Available fonts - these can be detected by rendering text to an off-screen canvas then hashing the resulting graphic.
• Anything available from their partner organisations which you accessed in the same Incognito session.
• Your interests and activity itself. If you see a link in normal mode, they may well know you saw it, and when, and with what device. If it is shortly after accessed in Incognito mode from the same IP address, they may reasonably (and probably correctly) conclude it is still you but in Incognito mode.

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.