How do I stop blocked messages from showing up on MacBook Air?

Hello

I blocked a person from my phone and his texts still went to the messages app on my MacBook Air (Sonoma 14.6.1). How can I change the default settings so this does not happen again?

Thank you.

PS - I think I figured it out - I just assumed the default settings on my phone and laptop would have automatically synced - guess not.

[Re-Titled by Moderator]