I am using QGIS Version 3.16.4-Hannover and Windows 10 but I also tested on Linux Mint 20.
I have a problem with several imported shapefiles from different sources. I do not know if I do something wrong or if it is simply not possible.
I wanted to add vector data from other sources to an existing QGIS-project that bases on a OpenStreetMap-XYZ-Layer (https://tile.openstreetmap.org/{z}/{x}/{y}.png). My project CRS is EPSG:3857 (WGS 84 / Pseudo-Mercator).
You can reproduce the problem by copying this sample of vector files from Euratlas: https://www.euratlas.net/tele/1700_vector_data_sample.zip
The file includes shape-files covering Switzerland/Savoy.
I added, for example, seas.shp (from the above mentioned ZIP) to my project. The file's CRS is strangely recognized as "ESRI:54004 - World_Mercator". After importing, the layer is misaligned, its offset is about 4 kilometres to north-east. See screenshot, blue layer.
I tried to correct this by manually switching the layers CRS to EPSG:3857. In result, the misalignment grew! Now, the layer's offset is about 20 kilometres to south-east. See screenshot, orange layer.
Here's the original seas.prj:
PROJCS["World_Mercator",GEOGCS["GCS_WGS_1984",DATUM["D_WGS_1984",SPHEROID["WGS_1984",6378137,298.257223563]],PRIMEM["Greenwich",0],UNIT["Degree",0.017453292519943295]],PROJECTION["Mercator"],PARAMETER["False_Easting",0],PARAMETER["False_Northing",0],PARAMETER["Central_Meridian",0],PARAMETER["Standard_Parallel_1",0],UNIT["Meter",1]]
What am I doing wrong?
Is there any way to get the vector layer aligned to my project in EPSG:3857, thus to the OpenStreetMap-Layer?
Or do I misunderstand something important?
Is it simply not possible?
If so, could you explain why?

I openened a new project, added seas.shp to the empty project. QGIS recognized "ESRI:54004 - World_Mercator". Then I added the OSM-basemap.
The effect is identical to the blue layer in my screenshot above.
Actually, the offset mentioned above (3 km to the north) fits to the Lake Zurich area. In other parts of the map, the offset is differing. I fear this is complicating the problem...
– Alrik Mar 18 '21 at 08:32Any chance you know a database for detailed Holy Roman Empire and German Confederation borders, mid-15th to mid-19th century?
– Alrik Mar 19 '21 at 21:56right click / layer CRS / set layer CRS) to EPSG:4326, see screenshot: https://i.stack.imgur.com/cIAaR.png (the project CRS is in 3857 to fit the OpenStreetMap basemap in the background) - this is one of the relatively rare cases where setting ("changing") the layer CRS makes sense, see here for details: https://gis.stackexchange.com/a/383437/88814 – Babel Mar 21 '21 at 11:53