Nivard Ovington provided clarification for the question –"Army Number, Service Number or Regimental Number – which is the correct term to use?"
Generally, service number or regimental number is used, although that can be inaccurate depending on period of service.
The following relates to the British Army –
• From 1920 Army number might be used but overall service number is probably the most used term. Until 1920 it would be a regimental number.
• Each regiment had its own numbering system and sequence, meaning several men could have the same number at the same time if in different regiments. It was possible for the same number to be reused in the same regiment for different men.
• If a man was in regiment A and was transferred to regiment B, he would be allocated a new number, as can often be seen on WW1 medal index cards, its not unusual to find a man that has two, three or more numbers as he moved around.
• That all changed in 1920 (1917 for Territorials) when new army numbers were allocated. Blocks of numbers were allocated to the various regiments and all men were given a new number. Once allocated a man would carry that same number regardless of his movements, for example if he left the army and rejoined a different regiment he would retain the same number previously given.
More on Regimental numbers here
The Canadian Army issued a regimental number to a man on joining up and he retained that number regardless
Officers of either Army did not have a service number until much later.