Hi
National Insurance numbers are not the same in Gibraltar as in the UK, you won't get a new one as such, but you will pay Social Insurance from your earnings.The National Insurance number you were issued in the Uk is yours for life but doesn't apply over here, once you get a job you will need to register at the Gibraltar Tax Office.
All Gibraltar law isn't the same as the UK. for example, you can legally buy and consume alcohol at 16 in Gib, the minimum wage is £5, 75p less than the UK min wage plus there of other law and policy differences.
Not sure if you can change your banking address, but there is a Barclays Bank in Gib, so you can always ask.