Yes, according to the Ahl al-Sunnah if the caliph is chosen by the people voluntarily by giving of allegiance, then he should belong to the tribe of Quraish, and this hadith is among the evidence cited for this law. The ahadith which evidence this stance include:
إن هذا الأمر في قريش، لا يعاديهم أحد إلا كبه الله على وجهه ما أقاموا الدين
This matter (of the caliphate) will remain with the Quraish, and none will rebel against them, but Allah will throw him down on his face as long as they stick to the rules and regulations of the religion (Islam).
لا يزال الأمر في قريش ما بقي منهم اثنان
This matter (caliphate) will remain with the Quraish even if only two of them were still existing.
قريش ولاة هذا الأمر
Quraish are to be in charge of this affair
— Musnad Ahmad
قريش ولاة الناس في الخير والشر إلى يوم القيامة
The Quraish are the leaders of the people, in the good and the bad, until the Day of Judgement.
— Jami at-Tirmidhi
All of these ahadith were quoted by the sahaba to negate claims made for caliphate being given to other than the Quraish.
This stance is held by the Hanafi, Malikis, Shafi'is, Hanbalis and Zahiris. And by the rest of the Ahl al-Sunnah - scholars have cited consensus on it.
The Khawarij and Muʿtazilah and some others are known to have disagreed with this madhab.
The evidence for the alternate stance includes:
اسمعوا وأطيعوا وإن استعمل عليكم عبد حبشي كأن رأسه زبيبة
Listen and obey, even if the one appointed over you is an Ethiopian slave with a head like a raisin.
As an Ethiopian slave would not from the tribe of Quraish. Counterarguments are: This hadith only emphasizes obedience to the ruler once he has been appointed by giving an exaggerated example, it does not mean that it is actually valid to appoint an Ethiopian slave as the ruler in the first place. Further the hadith can also be interpreted to be specific to a governor or commander rather than the caliph, as the caliph is the one who appoints such sub-authorities over the people and these can belong to other than the Quraish.
قال عمر رضي الله عنه لو أدركني أحد رجلين ثم جعلت هذا الأمر إليه لوثقت به سالم مولى أبي حذيفة وأبو عبيدة بن الجراح
Umar said: If one of the two men were still alive, I would have referred to this matter to him and I would have trusted him: Salim the freed slave of Abu Hudhaifah and Abu 'Ubaidah bin al-Jarrah.
— Musnad Ahmad
As Salim was a freed slave and did not belong to the Quraish. The counterarguments to this report are: This report is weak. And if it is accepted to be correct then it is the madhab of a single sahabi which is not evidence near various madhabs, especially when it contradicts stronger sources of evidence. It is also specifically about Salim and that does not necessitate that it is a general law which can be extended to others. It is also possible that Umar held that Salim was eligible to be a caliph by considering him to be from the Quraish, as according to a hadith a freed slave belongs to the tribe of his previous owners - and Salim is considered the freed slave of Abu Hudhayfa ibn Utba who belonged to the Quraish.
مولى القوم من أنفسهم
The freed slave belongs to the people who have freed him
هذه الأحاديث وأشباهها دليل ظاهر أن الخلافة مختصة بقريش لا يجوز عقدها لأحد من غيرهم، وعلى هذا انعقد الإجماع في زمن الصحابة، فكذلك بعدهم، ومن خالف فيه من أهل البدع، أو عرض بخلاف من غيرهم: فهو محجوج بإجماع الصحابة والتابعين فمن بعدهم، وبالأحاديث الصحيحة
— Sharah Nawawi