The original question was "why does she sign with her maiden name"?

Whatever this may be (recorded in the Notice of marriage), according to Schedule C of the 1836 Registration Act, the names that appear in the Column "Name and Surname" in the Section headed "Marriages solemnized at the Parish Church &c, &c …" on the registration document must be the names that appear against the declaration "This marriage was solemnized between us ……...".