Whether its for the Covid-19 or the flu jab, the NHS Number is how the NHS identifies you, and the evidence of the jabs are placed in your medical record (including on the NHS App).
In some countries (eg France and Ireland) you cannot get into places like hotels, pubs, museums and restaurants without evidence of up-to-date Covid-19 vaccination. So having this evidence to show on your NHS app is very helpful.
More on the NHS number on MyHSN here. More on NHS app here.