All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad, sallallaahu ‘alayhi wa sallam, is His Slave and Messenger.
In Islam, the way to make a promise or conclude a contract is to shake hands. Ibn ‘Aashoor said,
“The pledge of allegiance (Bay‘ah) was done by shaking hands. The people would put their hands in the hand of the ruler when they gave the pledge of allegiance, as evidenced by the fact that when the Muslims pledged allegiance to the Prophet, sallallaahu ‘alayhi wa sallam, ‘Umar took the Prophet's hand and placed it in people's hands so that he, sallallaahu ‘alayhi wa sallam, would not be fatigued from moving it around to the great number of Muslims who were giving their pledge. This indicates that the people who pledged allegiance to the Prophet, sallallaahu ‘alayhi wa sallam, took hold of his hand.” [Abridged]
As for the "pinky promise", we do not know of any reference to it in the history of Islam, and in fact, it was borrowed from other cultures.
Perhaps the most relevant feature of Islam related to the "pinky promise" is the report that it is prohibited to imitate the greetings of the Jews and Christians. It was narrated on the authority of ‘Amr ibn Shu‘ayb from his father from his grandfather that the Messenger of Allaah, sallallaahu ‘alayhi wa sallam, said: “Whoever imitates other than us is not one of us. Do not imitate the Jews or the Christians, for indeed the greeting of the Jews is to gesture with the fingers, and the greeting of the Christians is to gesture with the hand.” [At-Tirmithi]
At-Tirmithi commented on the Hadeeth saying: “The chain of narration of this Hadeeth is weak. Ibn Al-Mubaarak narrated this Hadeeth on the authority of Ibn Lahee‘ah but he did not attribute it to the Prophet.” [End of quote]
This Hadeeth is supported by another report that An-Nasaa’i presented in his book ‘Amal Al-Yawm wal-Laylah on the authority of Jaabir . Ibn Hajar said in his book Fat-h Al-Baari that its chain of narration is good. But despite the explicit statement on the prohibition, some scholars held that it is disliked and not actually prohibited. Al-Munaawi said: "It is disliked to greet by gesturing and not prohibited as An-Nawawi has explicitly stated." [End of quote]
Therefore, it can be said that joining pinkies when concluding a pact is not prohibited in Islam.
However, it may be deemed prohibited given the established prohibition of imitating non-Muslims; however, the prohibited imitation of non-Muslims pertains to what is peculiar to them, but the practices or acts that are common among Muslims and non-Muslims do not fall into this category even if they were originally practiced by non-Muslims. If the "pinky promise" is not peculiar to the non-Muslims, then it is not a form of the prohibited type of imitating of non-Muslims.
At any rate, it is no doubt better and more appropriate to adopt the Islamic practice of shaking hands when making a promise or concluding a contract.
Allaah Knows best.