Hey meet51,
We're glad you thought to reach out to Apple Support Communities about the notification that appears in the App Switcher when a Bluetooth device is connected. You're able to completely turn off Siri Suggestions by tapping Settings > Siri & Search. Scroll down to "Siri Suggestions" to turn this off. You'll then want to Restart your iPhone for the changes to take effect. Do you still show the notification on the App Switcher after turning those off?
If you prefer to leave Siri Suggestions on, you're also able to adjust this within the app settings too. In the screenshot it shows Instagram, so you would tap Settings > Instagram > Siri & Search, then turn off "Learn from this App".
More information on Siri Suggestions can be found in the following guides:
Siri Suggestions, Search & Privacy
About Siri Suggestions on iPhone
Keep us posted.
Take care.