In TileMill the "Earth" was automatically displayed "continuously", which means there was no "end". I tried to find this option in QGIS now, but nor search not checking through the options, etc. brought any success
That's not possible in QGIS. The reason is related to projections, TileMill only supports Web Mercator projection, which preserves north/south. Up is always north, down is always south.
This is required to allow the wrapping that you see in TileMill or Google Maps.
QGIS supports all map projections, a very large number of which don't preserve north/south alignment so it's not really practical.