I think that is the correct behavior if your daily note folder and your default new note folder are different.
I have a daily note folder and my daily notes get created by the Daily Note plugin within that folder.
I have a default New Note folder, that is a different folder, where any new note is created. My daily note is not created in this folder.
When I click on my next day link in my daily note (I have a similar setup as you with the dates at the top), then this new ‘daily note’ is created in the new note folder and it uses my new note template, not my daily note template.
Basically, the Daily Note plugin isn’t watching for you to click on a non-created note link within your daily note and then grab that action and create the new note as a daily note. Obsidian just creates a default new note in the default new note folder when you click on any non-created note link.