Do I have the right to find out how the TV licence inspector found out my name?

Electoral register ?

You have a right to request all data that An Post have on you (with some exceptions): https://www.anpost.com/Privacy/Access-Request-Policy

Of course, that doesn't mean that An Post still have details on how they obtained that information. Nor does it mean that they are required to retain that information to proceed with a prosecution

If you send in a request will you have given them your name and address so they can reply? 😁

I do not believe they use postal items to garner your name. The Postal Warrants would not permit that

They can however use multiple data sources. An Post own a subsidiary company called Data Ireland. They may use various sources to build databases for sale to marketing and advertising companies. An Post TV licence section. may also buy this for detection of unregistered addresses. They may claim a legitimate interest in processing your data.

I would be more concerned about the consequences of lack of TV licence than the GDPR aspect.

Edit; I recall reading before they may get access to those signed up to sat or cable provider).

https://www.newstalk.com/news/an-post-could-get-access-to-tv-subscriber-information-in-license-fee-crackdown-583665

The Yes/No for commercial use is gone; its all No all the time now. Register can only be used for electoral purposes, the Edited Register that had other use is no more

Is there a TV in the house, no matter who the owner is?

Law has changed here too, its no longer about whether or not you own a TV, but if you are able to watch broadcast TV on, for example, a monitor, or have a TV aerial and satellite dish at the premises, its possible you might be perceived as "needing" a license. If you don't own a computer or have an aerial or satellite or ground digital connection, you can certainly fill out the forms and declare yourself to be entirely TV broadcast free.

General TV Licence FAQ's