No, The Platinum Card® from American Express is not an invite-only card. Anyone can apply for the Amex Platinum card, but approval will depend on your income and credit score.
You will need at least a credit score of 670 and an income of more than $50,000 per year. For the best chance of approval, you should have a 700+ credit score and a $75,000 income.
American Express has two invite-only cards, the Centurion and the Centurion Business, which require at least $250,000 in spending on Amex cards each year.