There's evidence to suggest that Lupo ran his own family rather than being a lieutenant of Morello. In a 1902 letter from Francesco Genova (of New Orleans) to Morello, he describes a young Sicilian as being in the "House of Lupo". "House of" was also a term used in Man of Honor when Bonanno recalls a discussion between Morello and Maranzano about the "House of Schiro" or the Schiro Family.
There are also other indications that Lupo held his own authority equal to Morello in New York at the time, such as in 1903 when an informant witnessed a meeting in a restaurant between Lupo, Morello, Vito Cascio Ferro and Morello's men. Lupo was the one leading that meeting. In 1909, Antonio Comito observed Lupo interrupting Morello, which was noticeably different from the behaviour observed from Morello's men. Just before Morello and Lupo were convicted of counterfeiting, Lupo was observed by Comito to berate Morello.
A deported mobster using the pseudonym Zio Trestelle also told author Giuseppe Selvaggi that Lupo promptly established himself as a boss shortly after arriving in New York in 1898.
It's likely the Family headed by Lupo was that of Nicola Taranto who was described as the "Supreme Head of the Mafia" by the New York Herald after he was arrested in 1896 for counterfeiting. Taranto's "right hand man" was Frank DiGregoli, who was from Palermo, like Lupo (unlike the Morellos, who were from Corleone).

In late 1913, the Secret Service learned from informant Salvatore Clement that there were four borgatas in New York at that time, lead by Manfredi Mineo, Nicolo Schiro, Fortunato Lo Monte (successor to Morello), and Salvatore D'Aquila.
Mineo and D'Aquila were both Palermitani, and both operated in similar areas in Manhattan and Brooklyn. It's possible the Lupo Family was split between the two of them or that one of them took over that Family and another group was formed. What's interesting is that when D'Aquila was killed, it was Mineo who took over that Family, meaning you would have had a boss transfer from one Family to another (which seems more likely if the two groups split off in the first place). By the late 1920s, one Palermitani Family goes from D'Aquila to Mineo to Frank Scalise to Vincent Mangano, and the other (likely the original Mineo Family before he transfered to the D'Aquila group) goes from Salvatore DiBella to Joseph Profaci in time for the Castellammarese War. Either way, you can see the origins of the Gambino and Colombo Families at that moment in the split in the Palermitani.
In the Morello Family, Lo Monte is killed in 1914 and eventually succeeded by Salvatore Loiacano. When Morello got out he tried to take control of the Family again, killing Loiacano and resulting in the Second Mafia War. You don't see evidence of the split in the Corleonesi until the 1920s, which likely was caused by Morello's war. Regardless, you see at that point the emergence of the Masseria-Morello Family and the Reina Family (though you can argue about which one split from which; after all, the Morello group were the rebels in the war). Masseria then has Reina killed and installs Joseph Pinzolo as the new boss (like how he replaced D'Aquila with Mineo), but the Tommy Gagliano group kills him and takes over, siding with Maranzano. Masseria and Morello are killed during the war and Charles Luciano takes over that Family.
For the origins of the Bonanno Family, you can trace them back to Paolo Orlando, then Sebastiano DiGaetano (who appears to have briefly been Capo Dei Capi during Morello's incarceration), then Schiro until 1930 and then Maranzano. Obviously, Maranzano is killed in 1931 and temporarily replaced by his underboss Angelo Caruso. Caruso then steps aside and Joseph Bonanno is elected boss.