Can someone else see my messages on his device?
You’re mixing up two different thing all happening on the back end of the messaging servers.
You receiving someone else’s messages doesn’t mean you can read their’s. You don’t know who got you old number and you don’t know who’s old number you got. The probability that the two of you just happened to swap phone numbers is highly, highly unlikely.
Is it possible that someone saw some of your messages prior to changing the number in iMessage? Yes. You don’t know who sent you messages intended for you to your old number.
the messages you got when you changed your number were likely queued up SMS messages that couldn’t be delivered until your number was activated.
is someone possibly seeing SMS messages sent to you at your old number? Yes. Apple has no control over SMS messages and has no idea if the sender meant to send it to you or the new owner.
Changing phone numbers is not a normal, everyday process. Just control the variables you can control and ensure your account information where your phone number is relevant is accurate. As far as someone having access to see all or even past messages, no, that’s not possible. Unless they have your iCloud account and can log into it, they can’t access anything.
Every Apple ID contains a list of phone numbers and email addresses that are linked to it. There is also a list of devices that are linked to the Apple ID, with encryption keys for each device.
When a "blue bubble" message is sent to a phone number, it checks for an Apple ID linked to that number, and then checks for encryption keys for each of the devices, and separately encrypts the message for each device.
Only the device it was encrypted for can read a "blue bubble" message. And Apple should be notifying you if a new device is added to your iCloud account (you can also log into appleid.com and check for your list of devices).
When a "green bubble" message is sent, which will automatically happen if a blue bubble fails, then all bets are off. The device can be read by anyone. There is some security, but it's not very good at all. And it also could be broken depending on how good your carrier is. The fact they've passed your phone number on to someone else without leaving it dormant for a very long time suggests they really don't care about this kind of problem.
If it was me, I'd be calling Apple and asking them to double check the old phone number isn't linked to my Apple ID. I'm not sure exactly when they deregister it, but it sounds like they haven't for you yet.