Hi Rosie
Welcome to Rootschat

As you say, the other topic you have found and linked to, seems to be your Grandfather - and confirms, he was born in 1903 (no father named on the certificate) in Scotland at Maryhill Maternity Hospital (Glasgow, Lanarkshire), a year and a half before his mother Bridget MOONEY married Michael McGRORY, in Maryhill.
The other topic further states Bridget and Michael both born Ireland.
If you post a message on the other topic, hopefully both the OP (original poster) who may be your cousin and the kind person who had the copy of the birth record, may see your post via a Rootschat notification and repply to you!
Kind regards
AMBLY