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No, you cannot do it like that. Google Drive works in a very weird way. Shared folders are not, in fact, folders on your drive as far as Google Drive is concerned. They are an entirely different thing which just happens to be named in a way that is confusing to every sane person (those of us who had to deal with Google's APIs have long lost our sanity…). The Google Drive API we are are using only allows us to upload to folders in a drive the user has access to, not Shared Folders.

However, you are approaching this a bit backwards.

You can have your client log into their Google Drive through Akeeba Backup's Configuration page and send you the access and refresh token (you really only need the refresh token, by the way). You can then enter the refresh token in the backup profile, click on Save, enter the name of the folder you want the backups saved to, and click on Save & Close.

You can still manage the backup files of your client's sites through that site's Manage Backups page. If you need to download a backup you can do so through Akeeba Backup's Manage Backups page Manage Remotely Stored Files button next to the backup record; it will download it back to the server and then you can download it to your computer using SFTP, FTP, or the hosting control panel's file manager. After you're done downloading go back to the Manage Backups, select the backup, and click on Delete Files. This only deletes the files stored on the web server (it does not touch the files on Google Drive). You can also use the feature to download the backup files to the web server if you want to perform a restoration.

Of course, if your client shares the Google Drive folder where the backups are stored with your Google account you can also see them in the Shared With Me section of your Google Drive web interface and download them directly, without going through Akeeba Backup. I just gave you all possibilities so you can decide which one fits better your particular use case and workflow. That's the whole point of giving a multitude of options to do the same thing. Different people use the software in very different ways. We try to cater for as many use cases as we reasonably can.