I was able to find some more records relating to Matthew Young from the British Royal Navy & Royal Marines Service And Pension Records, 1704-1919, currently free to access at Find My Past.
His name appears in the list of examinations on 26th March 1817
His name appears in the list of examinations on 18th December 1827
His name appears in the list of examinations on 24th November 1834
His name also appears in a list with the word ‘pensioned’ in November 1834 at the age of 42 after 20 years service (meaning he joined the Royal Navy around 1814.
These records don’t quite tie up with his career as published in the British Naval Biographical Dictionary, 1849, so I will need to do more research.