I've come across records before where the father's name is not recorded, but this is the first time I've noticed it for the same family for multiple records. I'm assuming it is the same mother in both cases, mainly based on the address and how it ties in somewhat with the 1911 census record (http://www.census.nationalarchives.ie/pages/1911/Westmeath/Mullingar_South_Urban/Austin_Friars_Street/886067/). Is it a bit unusual or common enough?
Margaret's marriage cert (https://civilrecords.irishgenealogy.ie/churchrecords/images/marriage_returns/marriages_1918/09704/5520157.pdf) states that James was her father but I can't seem to find any other record mentioning James. And I can't find anything concrete in the 1901 census either.
You should try the parish for the baptisms to see what they say.
Genealogy Forum Mod
James Murphy the fruiterer married Catherine Lynch on 26th June 1881 in Mullingar RC; Catherine was aged 18, domestic servant, living in Mullingar, daughter of Patrick Lynch labourer https://civilrecords.irishgenealogy.ie/churchrecords/images/marriage_returns/marriages_1883/10951/8002555.pdf
In 1901 James & Kate were living in Austin Friars St Mullingar with two children 'boarders'; they had no children of their own. I would suggest that they fostered Bridget's children, both born out of wedlock. Margaret probably wanted to conceal the fact that her father's name was unknown and gave James Lynch as a good guess.
I checked the Mullingar records on registers.nli.ie but they don't seem to have anything for the dates I'm looking at (November 1893 and November 1903).
Thanks. It looks like Margaret was brought up by her aunt Catherine. Bridget married in 1895 and was listed as a spinster at the time. So Margaret was born out of wedlock like you said (born in 1893). Her first husband died in 1902 and Bridget married again in 1905. And that explains why there's no father listed for Patrick either (born in 1903). Looks like I might have reached a dead end with this side of the family without knowing who the father is.
And I think you're right about the name James Lynch. It appears it was just made up as her mother Bridget wasn't married at the time so her maiden name is Lynch.
I should have been clearer: contact the actual parish.