Who is a Brahmana

In the Katha Upanishad, one of the earlier upanishads...when the boy Nachiketas visits the abode of Yamaraaja, he is referred to as below.

"A Brahmin visiting your house is like Vaishwaanara Agni. Give him Udakam to propitiate him, else he will burn everything that is worthwhile to you".

Now the question is - where is it mentioned that Nachiketas was a Brahmin? His father conducted a Yaaga but that's not the preserve of only Brahmins of that age. Somewhere I read Swami Vivekananda actually referring to Nachiketas as the son of a King.

The one thing that distinguishes Nachiketas was his Sraddha, ie Aasthikya Bodha. It is this Sraddha that drives Nachiketas to seek the meaning of Death. This Sraddha makes him ignore all the material pleasures offered by Yama Raaja and make him ultimately eligible for Brahma Jnaana.

I feel that having Sraddha, the Aasthikya Bodha is what makes one a Brahmin.