The black stone itself is kissed. Some Muslims take this to be an important aspect of Hajj. Given the huge crowds during Hajj, however, it becomes difficult for people to get to the black stone and actually kiss it. Thus, one finds Muslim "how-to" guides for getting to the stone (like this one: "Pucker Up! How To Kiss the Black Stone Every Time")
This may remind one of the superstitious idolatry found in Roman Catholicism and the kissing of the toe of the statue of Peter in Rome. But it gets worse. The Hadith provide the following comments on the black stone:
It was narrated that Ibn ‘Abbaas said: The Messenger of Allaah (peace and blessings of Allaah be upon him) said concerning the Stone: “By Allaah, Allaah will bring it forth on the Day of Resurrection, and it will have two eyes with which it will see and a tongue with which it will speak, and it will testify in favour of those who touched it in sincerity.”And again:
Narrated by al-Tirmidhi, 961; Ibn Maajah, 2944
This hadeeth was classed as hasan by al-Tirmidhi, and as qawiy by al-Haafiz ibn Hajar in Fath al-Baari, 3/462
It was narrated that Ibn ‘Umar said: I heard the Messenger of Allaah (peace and blessings of Allaah be upon him) say: “Touching them both [the Black Stone and al-Rukn al-Yamani] is an expiation for sins.”Finally, the somewhat contradictory testimony:
(Narrated by al-Tirmidhi, 959. This hadeeth was classed as hasan by al-Tirmidhi and as saheeh by al-Haakim (1/664). Al-Dhahabi agreed with him).
Narrated 'Abis bin Rabia:And I'll just leave it at that. There is an odd religious veneration of the black stone in Islam, though they do not consider it a "god" yet they give it an extraordinary amount of attention.
'Umar came near the Black Stone and kissed it and said "No doubt, I know that you are a stone and can neither benefit anyone nor harm anyone. Had I not seen Allah's Apostle kissing you I would not have kissed you."