Yes, the Kaaba that is in Hajj is the House of Allah. Some cultures may not agree but Muslims believe it is.
Kaaba
Up to 4 million Muslims can perform Hajj and can be afforded in Al-Masjid Al-Haram that is surrounding Kaaba.
TawafaKaaba
Hajj or Islamic Pilgrimage.
Hajj always takes place in the Kaaba, located in Mecca, Saudi Arabia.
KAaba
Stones to throw at kaaba
Hajj four facts Where does Muslims do Hajj? Muslims pray in the Kaaba. Why does Muslims do Hajj? Once, Mohamed sad to Muslims, that third best think after belief to Allah is excellent Hajj. What are they doing when they get there? They are praying & praise Allah-it is calling Tawaf
Hajj is the fifth pillar of Islam five pillars. Hajj is pilgrimage to the Allah (God in English) House (called Kaaba) in Makkah (or Mecca) in the Kingdom of Saudi Arabia. Refer to link below for hajj ritual worship.
Hajj is based in Mecca (Makkah) because Hajj rituals are according to the prophet Abraham practices in Mecca and its surroundings. Abraham raised the foundations of Kaaba the oldest worship House of Allah. Turning around Kaaba for praying dates back to these ages. Many other Hajj rituals are in memorization of the prophet Abraham and his wife Hajar and his son the prophet Ismael.
According to Islam, it was Prophet Adam, seeing as he was the first builder of the Kaaba.