Guru Gobind Singh Ji gave every initiated man the name Singh, lion, at the creation of the Khalsa in 1699, though Rajputs had used Singh before. Tradition says he gave every woman the name Kaur on the same occasion, but the early Sikh texts do not show it, and the one early text that sets a naming rule for women says Devi. Kaur became general for Khalsa women only in the Singh Sabha period. It is usually glossed as princess and its etymology is argued.
The point was the removal of the caste surname. A name in Punjab announced caste, and a shared name removed the announcement.
It did not work. Caste names came back as a third element, after Singh and Kaur, and are now standard: Sidhu, Sandhu, Gill, Brar, Dhaliwal, Bains, Kalsi, Chahal. Village names are used the same way. A Punjabi name today usually states caste as clearly as it did in 1698.
Kaur is also, unusually, a name a woman keeps. It is not her father's and does not become her husband's, which in a patrilineal society is a real thing and is rarely noted as one.
The gap between what the name was for and what it is now used for is the shortest available summary of the position of caste in the Sikh Panth.
Singh and Kaur were given at the amrit sanchar of 1699 to remove the caste name from the personal name, which is a direct attack on the thing a surname does.
It did not work as intended. The got came back as a second surname in most families, and Singh became a middle name rather than the name it replaced.
Kaur has no equivalent anywhere. A woman keeping her own name at marriage, prescribed three centuries ago, is a practice a great many Punjabi families still do not follow.