Theoretically it is possible, but in reality it will never happen. There is certainly no mechanism or database that wallets or exchanges use to check if an address has already been used when generating new private keys.

The reason it will never happen is simply down to math. The numbers we are dealing with here are unimaginably large. For example, if every human on the planet each generated 1 million new addresses every second, and had been doing so since the birth if the universe 13.7 billion years ago, we would only have generated approximately 0.0000000000002% of all possible addresses.