Details
When updating an existing page on a destination site that uses taxonomy-based navigation with friendly URLs, the destination term will be deleted. This only happens when the page already exists at the destination and the version history is kept, due to the way ShareGate Migrate preserves it. The destination item must be deleted and a new item is then created with the source's history. Unfortunately, SharePoint removes the terms as soon as the page is deleted, seeing it as unused.
Workaround
There are two ways to avoid this issue:
Copy the page without the version history. This will only update the latest version to match the source changes and the navigation will be preserved.
Deactivate the navigation by taxonomy prior to copying the page and reactivate it once the copy is completed. The whole history can then be copied without problems. Since the friendly URL configuration is set directly to the term, the navigation will work as before, once it is reactivated.