New Macbook pro M1 pro warranty expire in 3 months ?

Nobody knows when you boot up a device for the first time except the person doing it. If it was wrapped in plastic, it should be fine. Just call Apple and ask to correct the purchase date/start of warranty since you just bought it from an authorized reseller. Be ready to give them serial number, purchase receipt copy if they want it, and whatever they ask for.

The 16 inch 2021 model MacBook Pro was discontinued last January.

I understand you bought it only last week, but even it has not been used before it must have been resting for a period of time measured in months, at least, and perhaps since last June.

In an abundance of optimism you might be able to convince Apple to update your purchase date but you must contact them by phone to do that. Refer to this excerpt from About the purchase date or expiration date for your AppleCare agreement - Apple Support:

If your coverage expiration date is incorrect

If the estimated expiration date of your Telephone Technical Support, Limited Warranty, or AppleCare agreement for your serial number is incorrect, please contact us. You will need to send the original sales receipt of your product to Apple so that we can update your purchase date. A sales receipt with the receipt number, product description, original date of purchase, price, and reseller details constitutes a valid proof of purchase.

Amazon is not an official or authorized reseller. Any third party marketplace vendor selling on Amazon would have purchased the product from somewhere, so in reality you are buying a pre-owned laptop even if it is in a brand new unopened box. The marketplace vendor is the original purchaser from Apple so the start date of coverage began at that point. Why buy from Amazon when you can buy direct from Apple or an authorized Apple reseller?

Also, be careful if a vendor offers to provide their own product warranty because that warranty will likely require you to use some specific unknown repair service which may not have any ties with Apple.

No, the laptop does not need to have been activated previously. Apple starts the warranty period when they ship the product to the reseller or first owner - that is how they have done it for many years. That means it was shipped and the warranty started in late June 2022. It does not mean it has been used before - it can simply mean that it sat on the shelf at Amazon since then. Usually, Apple will you allow to correct the purchase date if you bought from an authorized reseller - I've done that a couple of times. Call Apple (800-275-2273) and tell them you got it sealed and brand new from Amazon and ask what you need to do to start your warranty now. You will need a purchase receipt for sure.

They do not know if/when it was activated since the warranty has already started. But,you should be able to tell: was the box still factory sealed? If you opened it and booted it up, how many charge cycles showed? Was there a user account?

... I also bought a hardshell case for it.

Return the hardshell case. Do not use cases, skins, etc. with Macs.