It happens too when I try to upload an image to be used as profile image

Problem solved. I had to add this directive for PHP to my subdomain configuration so it could find its temporary files directory :

upload_tmp_dir=/var/www/vhosts/thamms.xyz/data_elgg/tmp

I've added the tmp directory with the right permissions in the data_elgg directory, and it works now.

Not sure why the automatic creation of the tmp folder would fail.

Anyway, the path /var/www/vhosts/thamms.xyz/ looks to me to be the document root folder of the domain, i.e. everything in it is likely to be accessible over the Internet if you haven't configured access to the data_elgg folder specifically. Creating the data directory inside the document root folder is not recommented. It's recommented to create the data directory on the server outside the document root folder to be on the safe side.