That kind of depends on with Whom your e-mail account is placed with ?
Running 4 Different Apple machines and 1 iPhone
3 of 4 on Ventura 13.2.1 and fourth stuck on High Sierra macOS 10.13.6.
iPhone on iOS 16.3.1
Q - " As the title say, the mail app doesn't sync automatically after the startup. I need to open the mail app to start receiving new mail. "
A - That behaviour it totally normal
A - The Mail Application in macOS for Apple computers must be Opened to Sync and New Messages received.
A -The behaviours of the Mail Application in macOS Does Not Act like Mail in iOS where the mail app is always Polling for New Message ( if configured to auto poll for new messages )