Safari opens in non-private window when clicking email links on Mac mini

I am using a MacMini with an M4 processor, macOS 15.6 (Sequoia) and Safari 18.6. In the Safari General settings I have Safari set to open with a new private window. This works as expected when I open Safari from the Dock icon but when I click on a link in an email, Safari opens in a non-private window. This only happens when Safari is not already open. When Safari is open, clicking on an email link opens a new private window. Is this the way it is supposed to work or is there a way to change this behavior?

[Re-Titled by Moderator]

Original Title: Safari is set to open with a new private window but it does not do that when I click on a link in an email.