The pair dated briefly last summer and new song 'The Smallest Man Who Ever Lived' appears to reference The 1975 frontman with lyrics including "Jehovah's Witness suit" and "rusting my (Swift's) sparkling summer".
Matty's aunt Debbie Dedes - who is the sister of his famous mother Denise Welch - told MailOnline: "Nothing surprises him any more. He will not be surprised by the song. Him and her know what went on.
"She writes about all her relationships, doesn't she? I don't think it will come as a shock to him at all.
"He's very happy in his new relationship so I'm sure he will be focusing on that. "As my nephew, we know a bit more about what went on than has been in the press."
Taylor said her new album describes a "fleeting and fatalistic moment in time - one that was both sensational and sorrowful in equal measure".
She added on Instagram: "This period of the author's life is now over, the chapter closed and boarded up.
"There is nothing to avenge, no scores to settle once wounds have healed. And upon further reflection, a good number of them turned out to be self-inflicted.
"This writer is of the firm belief that our tears become holy in the form of ink on a page. Once we have spoken our saddest story, we can be free of it.
"And then all that's left behind is the tortured poetry."
Taylor, 34, and Matty, 35, had a brief relationship last year, which ended in June.
He is currently dating 26-year-old model Gabbriette Bechte, while Taylor has moved on with NFL star Travis Kelce, 34.