Hi!
The reason why the plugin is not visible on other pages is because the shortcode is used only on that one page. The plugin cannot automatically place itself on pages, wordpress does not allow that yet.
I can recommend 2 possible solutions:
1. Use the Ajax search pro widget in the footer. Your theme most likely have footer widgetized areas. These widgetized areas are visible on the whole website usually. Use the search widget on one of these areas and it should be visible on the whole site.
2. Placing the search PHP shortcode directly to the active themes footer.php file. This solution works better, but it requires a bit of programming knowledge. On the demo page the compact shortcode is placed into the theme footer.php file just before the closing body tag: https://i.imgur.com/k7pzeXJ.png
In either cases, don’t forget to clear the cache and remove the shortcode from the baumkind.com/suche page editor, as it will be no longer required there.