Email notification and the like usually don't work out of the box and require SMTP configuration. This is usually available in the app admin consoles or configurable through Environment variables in the console.
I'm not running it ( and I haven't checked it for Monica HQ ), but those would be the first things I'd look for.
The .env.example file shows SMTP settings. The example file notifies to a logfile rather than an email.
https://github.com/monicahq/monica/blob/main/.env.example
# Mailing
# Set the MAIL_MAILER to 'smtp' if you want to send emails.
MAIL_MAILER=log
MAIL_HOST=smtp.mailtrap.io
MAIL_PORT=2525
MAIL_USERNAME=null
MAIL_PASSWORD=null
MAIL_ENCRYPTION=null
MAIL_FROM_ADDRESS=hello@example.com
MAIL_FROM_NAME="${APP_NAME}"
MAIL_REPLY_TO_ADDRESS=hello@example.com
MAIL_REPLY_TO_NAME="${APP_NAME}"