Best I can base my info on is Morgan Murphy's death certificate from Chicago, IL USA in 1932, where his son Matthew J (Joseph) Murphy as informant stated that Morgan's parents were Matthew Murphy and Mary Gaffney of Co. Wexford, Ireland. (attached here).
The death certificate is also where the birthdate Oct 20, 1866 comes from, although I'm not as confident about that date, as it "floats" around from 1858 (one instance on a family tree done by a descendant), to 1862, 1863, 1864 on the various years census. His marriage certificate from 1888 (also in Chicgao) implies 1863 (age 25 in Feb 1888), which perhaps fits better in the family "line".
It is curious to me that I don't find Morgan's baptism, when I do find so many (but not all) of the other siblings...so your thought that Morgan is from another family is entirely plausible. The Morgan and Matthew names do run down the tree...to the current generations, for what that's worth.
My Morgan and Bridget were definitely in Chicago before well 1901, Morgan marries 1888, Bridget marries Edward Cavanaugh in 1887. Morgan does his first papers for U.S. citizenship in 1886. Sadly, they didn't collect much information at that time.
Here are some other tidbits that may raise an eyebrow for you:
---1910 census in Chicago, the family next door is Nicholas Doyle, age 35, born in IL (with family). Do you have any "runs" of the Nicholas name?
---1930 census in Chicago, a "nephiew" Mathiew Murphy, age 36, born in Irish Free State, Caretaker/Laborer in a cemetery is listed. Says immigrated to us in 1927
Also in household is Furlong, John, listed as "nephiew", rents for $40, age 40, married at 30, b. Irish Free State, Baggageman Railroad. Came to US 1925, Hannah, wife, age 41, married at 31, born Irish Free State, came to US 1925, William, son, age 10, born in Illinois. (interesting note, if son is 10 years old and b. in Illinois, how could Mom have not come to the U.S. until 1925?)
I can share these census pages with you, but I'll have to dig them out. The above statements just come from the notes page in my Legacy Family Tree software file.
I've been away from this family line for some time...I'll have to get back to looking for immigration records (my theory that Morgan and Bridget came to the U.S. together about 1885-86?).
I'm happy to share the documents that I have on the Cavanaughs here in Chicago.
I appreciate your work on this---thank you.
B.