Yes, you can have both a UK-issued AmEx and a U.S.-issued AmEx. I've retained my Platinum UK-issued IDC card while also having a U.S. Centurion.
As stated by others, you will need a U.S. address and SSN for the U.S. issued card, but they might issue it immediately if you just tell them you are already a UK AmEx cardholder. Instead of doing a traditional U.S. credit check for you, they might just confirm your details with AmEx UK.
I would also consider adding the Platinum IDC card to your account, issued in the UK. There will be no problem with AmEx UK in allowing you to retain a card in sterling and one in dollars. The Platinum IDC carries the same good insurance as your current UK card (unlike the U.S. platinum card), and also provides two Priority Pass accounts (none for the U.S. platinum). I'm not sure if you already have Priority Pass with your UK sterlng platinum. The MR earning rate on the IDC is 1 point per US$1 spent.
Be careful on the medical insurance, btw. You are only covered when traveling outside your country of residence. So if you are considered in any way of resident of the U.S. during this period, you wouldn't be covered. Also, a trip is covered up to four months only, so if you are in the U.S. for any single trip exceeding four months, the coverage would cease.
Susan