For income tax purposes, it doesn't matter how many jobs you have, you are only entitled to one personal allowance in any tax year. National Insurance works in a different way.
Usually, when you start a job, you give your employer a P45 document. You receive this when you leave a previous job.
If you start another job without giving up your first job you won’t have a P45. Instead your new employer will give you a P46. This form tells your new employer you are working somewhere else as well. You don't have to tell them where you are working or how much you are earning.