How to prevent an app from sharing its data with Google? The changelog of a popular SMS/MMS app, Textra, states:

Textra is now backed up automatically [emphasis added] using the built-in Google Backup service. Makes it so easy to swap devices or uninstall / reinstall without losing your settings and any customization! Supported from Android 6.

Source: https://web.archive.org/web/20210519012025/https://play.google.com/store/apps/details?id=com.textra
What does this really mean?  In Android 7.0 is there a way to disable a single app from sharing private/personal data with Google?  If not, how about all apps?
The app itself does not appear to have any options regarding this unwanted functionality.
See Data backup overview. Since changelog is talking of Android 6 compatibility, it is referring to Android auto backup (see table).

Enabled by default. Apps can opt out by disabling backups.

Contextually, it implies Textra had opted out of back up earlier and now has it enabled.
There is no option for the user to choose which apps can be backed up or not. Either they backup or not (search settings for backup to see the toggle switch). If backup is enabled, apps that are enabled for backup are backed up.
Developers can choose to exclude some parts of app data even if backup is enabled. Backup data size is limited to 25 MB and doesn't count for your drive storage usage. Users have no control on what is backed up, so if you don't trust Google (good idea) look for custom ROM options as suggested by Izzy in comments or turn  off backup in settings.