whilst the above is probably correct that the burden of proof is on them to prove that it is you, this isnt a scientific proof, or courtroom stuff it is about making your life easier.
I had a problem some years back where a debt collection agency had found me and would not accept that they had the wrong person. According to them I had taken out a loan when I was a year old and they would contact me through various means most months for a couple of years or more. I would regularly speak to them and they would check the file. I used to refuse to answer the security questions and tell them just to look at the file- they would threaten not to discuss it with me!!! Capquest, absolute shambles.
I spoke to supervisors and managers, 2 months later the same repeated. I still get the odd letter but it is down to 1 a year or so, maybe less. They will chase from time to time to keep it alive.
if it is a genuine place- a debt collection agency that you can google and check, and they have traced a debt to you incorrectly, I would take a few calls and tell them. give them a chance. If they persist, then threaten them with a solicitors letter for harassment. it may have an effect. If it still persists beyond that, I can find you a template letter (although this isnt my area of law, I am sure that we can work it out!).
The company probably have limited info- balance outstanding, date of birth, last known addresses of the person. Your proof may only be to prove that your date of birth is different and it should remove the link to you. The burden isnt on you, but it makes life easier to get rid of it being associated with you.
get a credit report, make sure that the debtor isnt being associated with you and causing you other issues. Again, I had to remove a number of incorrect traces when I first did mine.
You could do nothing, and probably eventually it may resolve itself, but I wouldn't want to risk bailiffs turning up (I have seen the hassle that innocent house-sharers have on "Cant pay we'll take it away") and I got sick of the letters, calls at random times, texts etc. Give them a chance but bite back if they start to take the water!