Mine's been rock solid with its connection to my iPhone XS running 13.1 (I have a developer profile so I've been on it since it was released to testers.
Only issue was with the original 13 update that forced all existing bluetooth connections to be re-confirmed. I unpaired the Echo from bluetooth settings as well as from the Alexa app and re-paired it once. It's been fine since.
FWIW, I've been running 13.1 as well. My XS shows 4 BT connections. Amazon sent me these steps to fix in the meantime... Perhaps they will help others here.
"
Hello,
Follow the below steps:
Force close and restart the Alexa app, wait 30 seconds and make a new request.
If that does not work, close the Alexa app and restart your iphone.
If still the above steps doesn't resolve the issue, please follow the below steps:
Unplug the power cable for 45 seconds from your Echo device.
Go to your phone's Bluetooth Settings > unpair/forget Echo Auto from paired devices list.
Plug back the power cable power to Echo Auto.
There will be a prompt starting with " Echo Auto should play TTS - "Hello! Let's get rolling...". If you do not hear the TTS or a sweeping orange light after 20 seconds, then hold down the action button for 8 seconds to put the device in setup mode. Device will say TTS "Now in Setup Mode..."
Open the Alexa app > Devices > Add Device > Amazon Echo > Echo Auto.
Follow the instructions in the Alexa App. "