When looking for my institution, I find several organisations (none of which have a correct affiliation name) - how are these named variants retrieved/managed?

You can perform the disambiguation of your organisation (de-duplicate multiple names, identify parent-child relationships) using the OpenOrgs service. Then, following the scheduled workflows of OpenAIRE's internal procedures, your edits become available in due time to the OpenAIRE Graph and to all OpenAIRE's services.