If you can't connect a Bluetooth accessory to your iPhone ...
iOS 13 has following improvements and nothing in specific to Bluetooth change has been highlighted by Apple in iPhone 11
You can now select Wi‑Fi networks and Bluetooth accessories right from Control Center.
PlayStation 4 controller support
Use your PlayStation 4 game controller with iPhone.
Xbox Wireless Controller with Bluetooth
Connect your Xbox Wireless Controller with Bluetooth to iPhone.
Find offline devices
Locate a missing device even if it’s not connected to Wi‑Fi or cellular using crowd‑sourced location. When you mark your device as missing and another Apple user’s device is nearby, it can detect your device’s Bluetooth signal and report its location to you. It’s completely anonymous and encrypted end‑to‑end, so everyone’s privacy is protected.
Bluetooth album art
iOS supports the AVRCP 1.6 standard, which includes album art when connected to a supported car kit or speaker.
Wi-Fi and Bluetooth location privacy enhancements
API changes and new controls help prevent apps from accessing your location without your consent while you’re using Wi-Fi and Bluetooth.