An ID thief would most likely buy whatever phone he thinks wouldn't set any alarm bells ringing in the minds of the shop staff (i.e. avoiding £500+ models) and which he thinks he can sell on easily on eBay or Gumtree. Remember that he might have obtained ID information for thousands of people and know that lots of small transactions can slip through the net, whereas a few really big ones might look suspicious.
Either way though, it's clear you name and address has incorrectly ended up on a list of debtors held by a perfectly reputable company. It's either a data entry error on their part (e.g. typing in an incorrect postcode, together with a house number, and not noticing that the resident's name doesn't match up with that of the debtor) or, still far more likely in my opinion, ID theft.
As it stands, a debt collection agency genuinely believes that you owe their client money, so you can't blame them if they keep on chasing you for it. That could (and almost certainly would) end up in court action if you simply kept ignoring their letters, so I still urge you to write to them in the way that I've suggested above.