When I open messages on my iPhone, it connects and shows a message popup on my Mac, even though I've done everything to disconnect it...

Why does a message icon pop up on my mac when I open messages on my phone? I turned off my message account on my Mac, but it still pops up when I open messages on my phone. Considering I don't want messages on my Mac, this can be very distracting. Can I turn this off but keep my AppleID on my Mac? Thank you so much!! :D