You did not state where Michael lived in the US. I found one reference to a Michael D. Castrigano, born in 1893, who was naturalized through his father, Vincenzo's naturalization in 1896 in Cleveland OH.
His father's naturalization petition does not list his wife or children, but the date matches the 10-24-1896 listed for his naturalization. Ohio, County Naturalization Records, 1800-1977; https://familysearch.org/ark:/61903/3:1:3QS7-L996-5YX7?cc=1987615&wc=4627-1DL%3A265566401%2C267236901
I could be wrong, but wouldn't Michael have automatically become an American when his father naturalized?
If this his possible 1893 birth in Anzi: https://www.familysearch.org/ark:/61903/3:1:3QS7-L94M-5QTF?i=2209,
Could this be his father Vincenzo's naturalisation in 1896: https://www.familysearch.org/ark:/61903/3:1:3QS7-L996-G9J6-4?i=32
Dear Ms. Rossi,
Thank you for posting your request on History Hub!
The National Archives holds naturalization records of individuals who filed naturalization paperwork in the Federal Court system. Prior to September 27, 1906, any "court of record" (municipal, county, state, or Federal) could grant United States citizenship. Often petitioners went to the court most geographically convenient for them. As a general rule, the National Archives does not have naturalization records created in state or local courts. However, a few indexes and records have been donated to the National Archives from counties, states, and local courts. Please contact the National Archives at Chicago (RM-CH) at firstname.lastname@example.org to determine if records from the lower courts in Ohio are available. Records from state and local courts are often at state archives, therefore we suggest that you contact the Ohio Historical Society Archives/Library for county records.
Due to the COVID-19 pandemic and pursuant to guidance received from the Office of Management and Budget (OMB), NARA has adjusted its normal operations to balance the need of completing its mission-critical work while also adhering to the recommended social distancing for the safety of NARA staff. As a result of this re-prioritization of activities, you may experience a delay in receiving an initial acknowledgement as well as a substantive response to your reference request from RM-CH. We apologize for this inconvenience and appreciate your understanding and patience.
Additionally, since Mr. Vincent Castrigano filed and received naturalization citizenship before Michael D. Castrigano reached the age of 21, Michael would have received derivative citizenship though his father. However, prior to 1906 when a standardized form was adopted by Immigration and Naturalization Service (INS) and the US Federal courts for the purpose of naturalization, the only information included in the petition was the individual’s name, current residence, date and place of birth (sometimes included port of entry and date of entry) but very rarely did it include the names of any dependents. After 1906 with the adoption of a standardized form, the name(s) and ages of children would appear on the form.
We hope this is helpful. Best of luck with your family research!