Voting, much like all other things in Estonia such as getting married/divorced, doing taxes, signing documents, starting/closing companies, notary dealings, bank dealings, selling/buying vehicles, and many more things I can't even think of right now are entirely done via the digital ID that every citizen has. This means that you authorize/sign actions with it, including voting, because only you have your private keys (either in your personal ID card, in your phone's sim card, etc) that you yourself know the PIN for, which then authenticates you as being you. I think we're now at a point where there isn't a single government or business dealing you can not do entirely online (https://e-estonia.com/solutions/).

> Can the PIN change?

You can change it in the app, yes.

> How to issue new key if needed?

I think you’ll have to reissue your ID.

There’s also digi-ID (similar e-signature certificate on a card, but without any ID features), Mobiil-ID (e-signature on a SIM-card, no idea how it works), Smart-ID (in app, tied to secure storage in Android/iOS, cross-signed by the server which is supposed to check the device somehow) and probably something else I don’t remember. All of these are independent options, so you can, for example, revoke your Mobiil-ID if you lose your phone, and still use the your main ID card to sign things.