Apple's iMessages does exactly this, as well as every other messaging app that's not SMS related. The Google Messages web app is seriously flawed and does not work as a main texting app for several reasons, with connectivity issues and sluggishness, lack of a search function and notifications not working being some of the issues. It can work as a backup on a tablet or computer I guess.Because of the inherent nature of messaging, most apps won't let you sync between two phones as the database with the native number would be in conflict. If you want to 'sync' messages on a second phone your only option is usually the web interface. Not sure why that wouldn't work on your device (especially on a Pixel), but I've had no issues with Google Messages (or Textra, back in the day) on a tablet and a couple computers via the web interface.