Thanks for your clarification.Whoever told you this is incorrect.
The firm will be paying tax and NI for you - the only way they won't be doing so if it is unpaid internship. Every job that pays employees in the UK will be subject to tax and NI - you are not exempt just because of the job title (as otherwise everyone's job title would be intern). You will need an NI number - they cannot employ/pay you without one.
Also, the firm are not looking at this from a visa perspective - they just think you can take leave to attend the vacation scheme. You need to check the terms of your visa carefully as you could be breaking the terms of it if you work for another employer who is not your visa sponsor. A Tier 2 visa is linked to your employer and their right to sponsor you, so going to other forms of employment is not just as simple as taking annual leave.
I am copying what is written in the gov.uk website, saying that we are allowed to get a second visa and do two jobs. I can be sponsored for the vacation scheme and I still have my current job.
All the Firms that I applied to are stating that they sponsor for Vacation Scheme. Thanks.
If you’re applying to add a second job to your current visa
You must apply to update your visa if you take on a second job that is either:- more than 20 paid hours a week in addition to the job you’re being sponsored for
- in a different occupation code
You’ll also need to include a letter with your application explaining that you want to change your current permission to stay.
Your letter must state:
- your name
- your date of birth
- your current certificate of sponsorship reference number
- the date when your current permission to stay runs out