The Rebbe addresses a concern about a shidduch where the bride’s father is named Avraham. He cites various halachic sources and affirms the ruling of the Tzemach Tzedek that since the names are not identical, there is no concern. However, he advises ensuring the father uses his full name when called to the Torah or signing documents.
https://www.torahrecordings.com/rebbe/004_igros_kodesh/shevat/901