A Caliph is an individual able to command the respect of the Muslim community. As such, the theological differences between Saudi's Salafi Islam and other Sunni sects created a non-trivial obstacle to reconciliation.

The Kingdom of Hejaz had the authority to make such a claim, given the legitimacy of the Hashemite dynasty. As you may be aware, they were the custodians of Mecca and Medina for a thousand years, and to this day have a reputation for good character. Jordan is still ruled by the Hashemite dynasty, and is known for accepting huge numbers of refugees from neighbouring Arab countries during times of crisis.

While it may be tempting to assume that the Muslim heartland is just Arabia, this is not true. Jerusalem is the third holiest city in Islam, a region Saudi Arabia never controlled. Jerusalem is regarded with great fondness by most Muslims, and the al-Aqsa Mosque and Dome of the Rock in the old city are sacred places. Similarly, many places in Syria and Egypt have great cultural and historical significance for Muslims. Damascus was the capital of previous Caliphates, and has long been a centre of Islamic learning.

To be recognised as a legitimate Caliph, in the traditional sense of unifying Muslims in their heartland, one really has to influence Palestine, Syria, and Arabia. Saudi has never done that, while preceding Caliphates have. The fractured state of the Middle East after the fall of the Ottoman Empire meant that nobody has been in a position to invoke the title with much seriousness.

One should understand that while the Hashemites claimed the title in the 1920s, during that era the region was in a state of flux. It was possible that a Hashemite dynasty could have assembled a successor state from the ashes of the Ottomans. But this did not happen.

During the First World War letters were exchanged between Hussein bin Ali, Sharif of Mecca, and Lieutenant Colonel Sir Henry McMahon, British High Commissioner to Egypt. The boundaries discussed would essentially have included most of Arabia, Iraq, Syria, and Palestine. In that case the Hashemite dynasty could have legitimately claimed to be a Caliphate. But the British government had made various promises to various parties in order to acquire as much support for the war as possible, and as such did not honour their negotiations with Hussein, who became King of Hejaz instead.

The Hashemites not only have Meccan pedigree, but they also briefly controlled East Jerusalem and the West Bank in Palestine in the aftermath of the first Arab-Israeli war.

Saudi Arabia could have claimed to be the Caliphate if their power extended to unify the Muslim heartland: inclusive of Palestine, Jordan, Syria, Iraq, Egypt. But though the House of Saud is undoubtedly super rich, they have never been powerful enough to force the issue. Even if they did, their dynasty does not have the legitimacy of the Hashemites.

In conclusion, many have claimed to be a Caliph, but in order to become the Caliph you must unify the Islamic heartland; which is far more than just Mecca and Medina. Saudi Arabia has never been able to do this, and has never claimed this as its intention either.

I answered a similar question several months ago here: https://www.reddit.com/r/AskHistorians/comments/df0f4p/is_there_a_reason_the_kingdom_of_saudi_arabia_has/?ref=share&ref_source=link

tl;dr: the Wahhabi movement considers the title of caliph "bida" (innovation) since it was introduced after the Prophet's lifetime (walking this back - I didn't get it from a neutral source, so I'm not willing to stand behind it -- there are other reasons laid out in this post); also, under the current government system, it's not clear that the Saudi King would be the person who held the title.

I am kinda late to the party, but I hope this answer gets seen. While the answer of u/khowaga has some good insights, I think there is still more left to be said about this question.

To answer this question we have to understand what the title of Caliph means, what are its requirements and what is its responsibilities.

To start, you need to understand that one of the core tenants of Islam is that all Muslims are brothers and sisters, they are members of one big entity called the "Ummah", the Ummah (literally: the nation) is a sociopolitical union, that is quiet different from the modern concept of the nation-state as it emerged in the 18th century Europe (previously defined by the UNESCO as: ".. an area where the cultural boundaries match up with the political boundaries"). We can summarize the main differences between the modern nation-state and the Ummah as follows:

1- Citizenship in the nation-state is almost always based either on blood/birth right (jus sanguinis) or place of birth (jus soli). Citizenship in the Muslim Ummah is based solely on religion, Islam, i.e. any Muslim in any place in the world has full rights of citizenship and full responsibilities of citizenship, no mater what their location, ethnicity, bloodline, place of birth, culture, language .. etc. are.

2- The Muslim Ummah is not bound by a specific geographic location nor is it bound by political borders. Any Muslim anywhere falls under the jurisdiction of the Ummah's leader and the Ummah itself tries to expand to add as much land to its political domain.

3- Any one who converts to Islam instantly gains all the political rights and obligations of a citizen (including monetary rights such as receiving protection and sustenance and obligations such as paying Zakat), and anyone leaving Islam surrenders all their rights instantly (such as the right of marriage to a Muslim, the right of inheritance .. etc.)

---

During the life of Prophet Muhammad (PBUH) he was both the supreme religious and political leader (in Arabic: Imam, literally: leader) of the Ummah. After his death, a Caliph (literally: successor) was to be chosen to take over the political duties of the Imam, but not the religious leadership. From this we can understand that the Muslim Caliph is an office that is the polar opposite of the Catholic Pope, i.e. he holds the political leadership on all Muslims everywhere in the world, but he has no religious authority over them.

---

Now there are specific conditions for a person to be selected to serve as Caliph (according to Sunni Islam), some of these are unanimously accepted and some are disputed, some of these are:

1- Islam: no non-Muslim is allowed to serve as Caliph, based on the verse 141 of chapter 4 of the Quran. " .. and never will Allah give the disbelievers over the believers a way [to overcome them]"

2- Manhood: no woman is allowed to assume the supreme leadership, based on the narrated Hadeeth from the prophet, commenting on Boran, the daughter of Khosrow II, assuming the leadership of the Persian empire: "Never will a people succeed if they chose a woman for their leader". (Narrated by Al-Bukhari in his Sahih, no. 7099)

3- Adulthood: must have reached puberty.

4- Sanity: must not suffer from a mental illness.

5- Ability: the Imam has to be able to do the jobs, which includes being brave, wise, able-bodied .. etc.

6- Piety and Islamic Knowledge: this is a disputed condition, the majority of Muslim scholars say that the Imam should be a pious person knowledgeable enough about Islam that he is able to formulate his own scholastic decisions in matters of state and religion (Mujtahid). While the Hanafi scholars say that this condition is a priority condition, but an impious person (Fasiq), or a person whose Islamic knowledge is not enough to be a Mujtahid can still be an Imam/Caliph.

7- Lineage: this is the most important condition for our discussion here, Prophet Muhammad (PBUH) said: "Imams (Leaders) are from Quraish, they have rights upon you and you have rights upon them. They should be merciful when asked for mercy, they should be just when asked for justice and they should fulfill their pledges. Whoever does not do that, then he shall be cursed by God, the Angels and all Mankind, and he will not be pardoned nor ransomed (in the day of judgement)" (Narrated by Al-Albani in Sahih Al-Jami, no. 2758).

Quraish is the tribe of the Prophet, and were the most prestigious Arabian tribe at his time, due to being the custodians of the Holy Mosque of Mecca. This is a disputed condition between the Muslim scholars, as some of them understood the phrase "Leaders are from Quraish" as a prophecy, foretelling what will happen, rather than a directive from the prophet. Others understood it as being merely a recommendation, based on the political situation in Arabia in that time. However, the vast majority of Muslim scholars agree that this is actually a required condition.

8- Election of the Muslim leaders: also known as (Bay'ah), this is an oath of allegiance that has to be given voluntarily from the Muslims (represented by their leaders) to the Imam, if his authority is to become accepted. This also has some requirements, such as: being in public, that he himself accepts the office, and that there is only one Imam given the Bay'ah at the same time and many others. Hereditary rule is strongly against the elective concept of Bay'ah.

---

For the first 5 Caliphs after the Prophet died, these conditions were met. However, the 5th Caliph (Muawiyah I) broke these traditions when he appointed his son Yazid I to be his successor. He forced the people to give the Bay'ah to his son during his life, thus establishing the first ruling dynasty (the Ummayads). For this reason, the vast majority of Muslim scholars regard Muawiyah is the first Muslim King rather than the 5th Caliph. This tradition was further broken because Yazid was very impious, known to drink alcohol publicly, among other things. However, the Ummayads retained the title of Caliph, and the unified political leadership of all Muslims, which was the same case for the Abbasids, until Hulagu Khan's destruction of Baghdad in 1258.

Now with the Ottomans things get a little complicated. The first Ottomans never claimed the title of Caliphs as they were missing a very important requirement, which is being of Quraish's lineage, since they were Turks rather than Arabs, instead they took the title of Sultan (which is a title for an autonomous monarch ruling under the nominal political authority of the Caliph). However, they started claiming the title of Caliph in 1517, albeit narrowly and scarcely, until the end of the Ottoman empire in the early twentieth century.

---

Now after this very long introduction, we can apply what we discussed on the Saudi Royal family to see whether they can qualify to being Caliphs. We will see that they are missing the two very important conditions 7 and 8. They are not descendants of Quraish and they have no wide-spread recognition by the Muslim world. Of course, many Muslims can argue that they are missing more of the requirements, but the rest of the requirements are difficult to disprove.

A very good book that I would recommend is (The Peace to End All Peace by David Fromkin) which discusses, in painstaking detail, the British attempt to create a new Muslim Caliphate in the early 20th century, by trying to establish wide-spread recognition to an Arabian ruler from the descendants of Quraish (Sharif Hussein, ruler of Mecca) to serve as a puppet through which they can unify the whole Arab world and use it to hurt the Ottomans. At the end, this turned out to be unnecessary as the Ottomans lost WW1 and the empire collapsed anyway, and the Arabs did not care a single bit about Sharif Hussein. However, this attempt ended up creating the ruling dynasty of modern day Jordan. The other ruling dynasty with a Quraish lineage currently is the dynasty ruling modern day Morocco, whose king sometimes claim the title "Amir Al-Muminin" (literally: Commander of the Believers), which is a title usually reserved to the supreme Imam/Caliph. However, most modern day Muslims view this as merely a joke.

Edit: Corrected some typos.