Linking Android to Chromebook Gnucash

Probably not. GNUcash is a software, not a cloud service.

At least the version of GNUcash I use (Linux) does not support anything except local file systems.

The closest to what you could have is to store the GNUcash file in Google Drive. At least the Android version supports that:

...but whether the Chromebook version supports that is anyone's guess. If you are running the mobile GNUcash (Android) application on Chromebook, perhaps then it could work. At least the information I found about GNUcash+Chromebook seems to be that it's the Linux version you use on Chromebook. Thus, it won't support remote file systems.

You could try something to mount Google Drive like a local file system. Not sure how well that works, though. In Linux it might work, but apparently not on Chromebook:

I am not sure which android app you are using, maybe "Gnucash Mobile for Android"? This seems to be no longermaintained.

Having said that, I would like to point out that it is possible to use Gnucash across multiple devices by using the SQL backend. It would require you to run a sql server such as Postgresql and possibly quite some cumbersome setup of all involved devices. But it should avoid the syncing problem of the file based approach.

The GnuCash Android app is not intended to automatically sync with your desktop GnuCash app. The developers explicitly state this on the GnuCash page on the Google Play Store:

NOTE: that the app does not offer full compatibility with GnuCash forthe desktop. You cannot synchronize between the desktop app and thisone. But you can import your accounts and transactions from GnuCashXML files.

I have GnuCash installed on both Windows 10 and Ubuntu machines. Those installations happily share (one at a time) a single data file stored on a mounted DropBox folder. I also use the Android version of the app as a mobile data capture system to enter details of my purchases while I'm out and about - so I can record details of my coffee purchase while I'm waiting for the barista to do her thing.

Once per week (or daily, if you prefer) the GnuCash app on my phone exports the new transactions to a folder on DropBox from where I can easily import the transactions into my main GnuCash system (using either Windows or Linux).

As far as I know, the answer to your question is No, you can't use either your Chromebook or your mobile phone to update the same GnuCash database with full synchronisation. But the mobile app is still a useful way to prepare your data as a file that can be easily imported into the main system.