If you’d like to customize the links that show up on the SharePoint Home site app bar, you can do so following the steps described in this post.
Here is how the app bar looks like on your SharePoint site:
Here is an example of how you can make your navigation look like when users click on the Home Site icon:
First, enable the home site using SharePoint Online Management Shell:
Connect-SPOService -Url ‘https://[tenant]-admin.sharepoint.com’
Set-SPOHomeSite -HomeSiteUrl https://[tenant].sharepoint.com/sites/Intranet
Then, follow these steps from Microsoft on how to configure the Home Site app bar