I'm not sure if you will accept this as an answer, but I will deflect your question on Tosafos by pointing out that this is not Tosafos' chidush. It is a direct quote from a censored gemara in Shabbos 104b which Tosafos in Chagiga referenced.
The missing gemara there says אמו מרים מגדלא.
Tosafos there says the people under discussion lived during the time of Rabi Akiva, so perpetuating the confusion and coincidences of the names and personalities under discussion.
בן סטדא בן פנדירא הוא אמר רב חסדא בעל — סטדא בועל — פנדירא בעל פפוס בן יהודה הוא אלא אֵימא אמו סטדא אמו מרים מגדלא נשיא הואי כדאמרי בפומדיתא סטת דא מבעלה:]
The Gemara asks: Why did they call him ben Setada, when he was the son of Pandeira? Rav Ḥisda says: Perhaps his mother’s husband, who acted as his father, was named Setada, but his mother’s paramour, who fathered this mamzer, was named Pandeira. The Gemara challenges: But his mother’s husband was Pappos ben Yehuda, not Setada. Rather, perhaps his mother was named Setada, and he was named ben Setada after her. The Gemara challenges: But his mother was Miriam, who braided women’s hair. The Gemara explains: That is not a contradiction; Setada was merely a nickname, as they say in Pumbedita: This one strayed [setat da] from her husband.