I know that you can add your own certificate to the domain and point that domain to the AWS Elastic Load Balancer. In my case I don't have domain, but would like still use secure HTTPS/SSL connection when talking client <-> backend. Is it possible to enable HTTPS connection directly to ELB, i.e instead of using http://some-random-url-here.eu-west-1.elb.amazonaws.com I would like to use https://some-random-url-here.eu-west-1.elb.amazonaws.com
That would mean, that AWS would need to provide the cert for the *.elb.amazonaws.com domain. I remember at least long time ago this was possible, but maybe my memory does not serve me right?
Memory does not serve you right. This is not possible now and would not have been possible in the past. ELBs don't have, and it is not possible to obtain, a certificate like this (including from Amazon Certificate Manager).
In fact, 3rd party providers like Let's Encrypt also have protections to prevent you from obtaining certificates like this, since amazonaws.com is not your domain.
You will need a domain that you control.