Have you found Matthew James in 1939 which will give his DoB?
For the benefit of others who may be following/helping, not much between the 2 births...
KENNEDY, MATTHEW JAMES STEPHENS
GRO Reference: 1889 D Quarter in CHEPSTOW Volume 11A Page 9
KENNEDY, MATTHEW JOHN POWELL
GRO Reference: 1887 J Quarter in CARDIFF Volume 11A Page 357
I'm intrigued, do you have a theory as to why he changed his surname, especially to such an uncommon name?
Could he have married & taken his wife's surname?I hadn't taken everything in, in one go

...was he possibly adopted?
Annie
Add...I had a look at the tree you mentioned, it appears the owner probably doesn't have the DC for the 1935 death as it's referenced as 'Mar 1935' with no specific date.
A few entries have specific dates, usually from National Probate Calendar, FAG & Soldiers died in the Great War