The solution that worked for @DannyDan I was still unable to log in. After some hours wasted, I found this:
# First, change the "login.service.js" file as proposed by
# jleifeld above.
cd /home/vagrant/shopware-dev/platform/src/Administration/Resources/app/administration
export PROJECT_ROOT="/home/vagrant/shopware-dev"
export ENV_FILE="$PROJECT_ROOT/.env"
export PORT="8080"
export HOST="192.168.33.10"
export APP_URL="http://192.168.33.10"
# Not necessary, personal recommendation:
export ESLINT_DISABLE=true
npm run dev
# ... wait until ready... then log in at
# "http://192.168.33.10:8080/"
Relevant data missing in a context … this seems to be a bit of a theme in development with Shopware 6 at the moment (not just in JS, unfortunately).
Since I am already writing this post: big compliment to the Shopware guys for the smooth installation of the vagrant environment 
! This is so nice, please don’t let it die!