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Hi Pey,
Thanks for posting back and providing the screenshots.
Regarding your query, it is because the two sites are different, one is a Team site, and another is a Communicate site. Communication sites only have the top(horizontal) navigation, while Teams sites do have horizontal or vertical navigation to choose from. It is a limitation by design.
Also, I tried to run PowerShell commands to set Vertical navigation bar for a Communication site, but it failed. Communicate sites don't have the property "VerticalQuickLaunch" at the backend. Sorry for the inconvenience that happened to you.
Based on the situation, I would like to suggest you vote for a suggestion in Microsoft Feedback Portal where other members voted for a similar suggestion here. It's the platform for us to hear your voice and improve our products.
Appreciate your understanding and patience.
Sincerely,
George | Microsoft Community Moderator