It's there to remind you how long you have been a customer of American Express. It's an emotional client retention tool.

In the USA the Member Since date is also the date that American Express reports to credit bureaus as the opening date for every card account in your name. This can have a beneficial affect on your credit score, and (to my knowledge) it's a practice not followed by any other issuer. I cannot say if the same practice is followed in the UK, nor even if UK credit scores give longevity the same weight.