Hello Sue, if the parents did not Marry, would the Child ( Catherine) still be given Her Father's Surname ? ( I have much too learn )

I did a search for a, Jane Cashin baptism, but only one on IGI is,

Jane Cashin 30 Jan. 1742 Maughold
death, 27 Dec. 1751
Parents, Wm. Cashin and Easther

this Jane Cashin died, would the Parents have named a later Child Jane ?

found a Marriage, Wm. Cashin and Ester Kneen 14 Oct. 1727 Maughold

burials,
Jane Cowle 27 Dec 1771 Mau.
Jane Cowle 25 Aug 1780 Mau.
Jane Cowle 11 June 1782 Mau.

burials,
Wm. Cashin 29 Aug 1769 Mau. 0106390
Wm. Cashin 8 April 1781 Mau. 0106390
Wm. Cashin 20 May 1783 Mau. ( no Will )
do you think these are the right Wills ?
is it by process of elimination ?
Averil