In my experience, the illeg male child is named for the (putative) father, with the mother's surname tacked on the end. Of course if there was an actual "relationship" which produced more than one child, the given first name would not always be the same as the father's.
The Parish Overseers would haul the pregnant woman up before them and demand to know the father of the expected child, so that he could make some payments for its maintenance, thus avoiding a cost to the parish. This would be recorded in a bastardy bond, and payments would be recorded in the parish rate books. Both of these still survive in some areas, and it is worth enquiring at the relevant records office in the hope that you will be lucky.
It was also likely that the child would then be put into an apprenticeship, (which have sometimes been indexed,) and one clue is if the child is rather young.... I have a 6 year old.
I do have one woman who most definitely had 2 illeg daughters; (the paperwork survives,) and they were both put into apprenticeships at the age of 8. Neither were given their respective father's surname. As the mother had an extremely unusual surname, I have my suspicions that there was a third illeg daughter, though sadly I do not have a paper trail to prove it..... as yet.
You ask if it is "strange to have two children with unknown fathers", a point that I have pondered at length, wondering if the mothers were bad or mad. Obviously some couples just got "carried away", and some women had only one trade they could use to make money.
In the case of the male child mentioned above, the father was a married tanner, (who probably smelled incredibly awful,) and I suspect the liaison may not have been the mother's choice!!