I could upgrade it successfully on a Ubuntu 20.04 with these steps:
# Backup data / you could avoid the param -t if you already have a proper influx config
$ influx backup /path/to/backup/directory -t [admin_admin]
# Back of data directories , just in case
$ sudo cp -a /var/lib/influxdb/.influxdbv2/ /path/to/backup/data_directory
# Stop service
$ sudo systemctl stop influxdb
# Remove previous version - Do not use param --purge
$ sudo dpkg -r influxdb
# Install new version
$ sudo dpkg -i influxdb2_2.0.3_amd64.deb
# Move data directories to new version location
$ sudo mv /var/lib/influxdb/.influxdbv2/engine /var/lib/influxdb/engine
$ sudo mv /var/lib/influxdb/.influxdbv2/influxd.bolt /var/lib/influxdb/influxd.bolt
# Start service
$ sudo systemctl sart influxdb