In order to facilitate the separation and the subsequent assignment to new columns you can use the extractall which returns rows with multiindex that can be easily rearranged in columns with unstack.
Input used as data.csv
Author_ID,Affiliation
6504356384,"Department of Cell and Developmental Biology, University of Michigan, Ann Arbor, Ml 48109, United States, Department of Computational Medicine and Bioinformatics, University of Michigan, Ann Arbor, Ml 48109, United States"
57194644787,"Department of Pathology and Immunology, Baylor College of Medicine, Houston, TX, United States, Texas Children's Microbiome Center, Texas Children's Hospital, Houston, TX, United States, Department of Pathology, Texas Children's Ho:"
57194687826,"Department of Biochemistry, Schulich School of Medicine and Dentistry, Western University, London, ON N6A 2C1, Canada, Department of Computer Science, Faculty of Science, Western University, London, ON N6A 2C1, Canada, Depart"
123456789,"Department of RegexTest, Address and Numbers, Department of RegexTest, Faculty of Patterns, Department of RegexTest, Department of RegexTest, City and Place"
import pandas as pd
df = pd.read_csv("data.csv")
print(df)
dept_names = df["Affiliation"].str.extractall(r"(Department of .*?),")
affdf = dept_names.unstack()[0].add_prefix("Affiliation")
affdf = affdf.rename_axis("", axis="columns")
affdf.insert(0, "Author_ID", df["Author_ID"])
print(affdf)
Output from affdf
Author_ID Affiliation0 Affiliation1 Affiliation2 Affiliation3
0 6504356384 Department of Cell an... Department of Computa... NaN NaN
1 57194644787 Department of Patholo... Department of Pathology NaN NaN
2 57194687826 Department of Biochem... Department of Compute... NaN NaN
3 123456789 Department of RegexTest Department of RegexTest Department of RegexTest Department of RegexTest