"I am thanking the customer."
Since there seems to be a large variety of words to choose for the word "customer" I decided to ask my German friend about them and when might be best to use each word. I'm copying and pasting what she said here in case anybody else was wondering too.
In a shop you would call a customer a Kunde, in a restaurant it's a Gast (guest) and if you're a freelancer it can be Kunde or simply the Auftraggeber (assignment giver). If you sell goods to a company, it's an Abnehmer (taker). In social work, law, psychology, ect, they have Klienten (clients). In a museum you would address them as your Besucher (guests).
Benutzer is an user (e.g. software user). Verbraucher is the whole group of people who consume things (consumers, also Konsumenten). This is mainly used e.g. in e.g. texts like "Verbraucher need to be careful when selecting a energy contract, says the ministry."
Käufer is simply "buyer". You can call someone in a shop a Käufer, but more professional nowadays is Kunde. It is a simple wording difference like in English as well (buyer vs. customer - which sounds better being used for the company).
Hope that helps!
Danke is one of these verbs that always leads to a dative object.
Essentially, danken is a "to-verb" in that it literally means "to give thanks to" which results in the direct object being dative.