In this case, Mr Coote was a British Soldier. We must keep in mind that the British army Balloned from from 100ks to multiple Millions. I am currently reading CALL TO ARMY - British Army 1914-1919. It highlights the BRITS were dealing Territorials, New Army and such. One key factor is there a was a major restructuring around 1912. The situation was there was no central control over the service / regimental numbers. The 1920s service number control was the result of lessons learned. Another thing to keep in mind is that everything was handraulic - easy to register not so easy to retrieve.
For Canadian Soldiers, I have seen on rare occasion as many as four (4) regimental / service numbers. A key observation is the Canadians retained the same service number when transferring from unit to unit. There are two circumstances that a Canadian would have multiple service numbers. One would be transferring from the NPAM (Non Permanent Active Militia) to a member of the CEF. The other situation is where an individual was released from service and rejoined the CEF, in that case he would be issued a new service number.
I just wonder if there was similar mechanism for the British Army soldier? It is one that I had thought about nor investigated why.