Not only is the Visa number the same, but so is the expiry and CVV number on the back of the card, so you can continue to use the same card info for your online purchases and if you've set up any PAPs with them. The new card product starts earning in the new rewards program by the next day, and in my testing with a new CBVI card I used the same card number info within hours after PS'ing and it already started to earn the 10% cash back.
The physical cards themselves though, will stop/start working once you call to activate the new card product. Which means you can even activate the new card before you receive it in the mail, but there's no advantage to do this as it cuts off your existing in-hand card. Also, if you already have this card in your Apple Pay it must be re-registered once you've activated the new card.