Hi - yes, it was usual in the IOM at that time and through the 1800s for the child to be known by the father's surname even if illegitimate. Although the IGI gives his/her surname as being the mother's one this is their own invention. In the parish register it would be written (in the ones I've seen) as John illegitimate son of Mary Bloggs and John Smith bapt .... (or vice versa), or as John Illeg son of Jane whomever, with no father (or child's surname) given. It helps with identification if an illeg child is with a grandparent in a census, or father's name and occupation are given at marriage (but sometimes invented), or is named in a family will. Or when the mother appears in a Presentment, with the father's name and residence given.

Sue