As sure as I can be without actually buying the certificates.
The profile has inconsistencies (mentioned already -- the birth date in the data field and the first part of the biography being not the same as that in the research note - the father's last name being not Shanahan/Shannahan) that have to make you look further. (The Walsh-Torr also made me look harder.)
Transcribers often get thing wrong, because handwritten records can be very difficult to decipher, so you will see things such as "May" when it was "Mary", or "Mary" when it was "May". (I have run across some pretty atrocious renditions of names - my own gr-gran is one such.) Unless you can afford to buy all the documents, or can get access to them by visiting the repository (which I used to do with my Mum when had access to Brisbane) you have to be able to put 2 and 2 together and be sure you get 4, not 7.
You will find many Irish women, in particular, will go by a Baptismal name, rather than a given name - or a confirmation name (usually a saint, or Mary). Then, too, are the use-names, such as your Gran using "Gay" - which has no apparent relationship to her given names (my Mum was known by a name in the WAAAF that was not, in any way, related to her given names), where keeping track of people is harder.
You don't have to take my word for it - in fact, I encourage you to check for yourself - but I spent several hours looking into it, double-checking (the Victorian BDM was down and I couldn't work on my line). If you want certainty, you need to already know the answer, already have documents, buy the record/s, or be/go to where the records are held to view them yourself. Otherwise it really is a case of ruling OUT what cannot be, until you have left over what may, possibly, be.
Even after I found proof my grX3grand was Irish, I kept checking for some proof -- any proof -- it was otherwise, because what I "knew" was that he was Scottish. Even my cousin - when doing the family history stuff as required by his church - did not find this, nor did he find my Swedish gr-grand's birth in Sweden (he, the cousin, still had him as England-born).
Research, eliminate, don't lock-in on family stories as they're not always accurate (although they may have grains of truth .. I confirmed a family "legend" by accepting the evidence I found and tracking it all the way).
I'm also not saying I can't be wrong, or that I never make mistakes, I can be very wrong .. and I have made mistakes along the way. But I work hard to correct those mistakes .. and I never present something as 100% sure, unless I am 100% sure. (I'll hedge it by saying "I'm not sure, but .. .." or "I may be wrong, but I found this .. .." and suchlike.)