QGIS 3.10.2. Projection Problem

Question

Updating QGIS to the new version I have serious projection problems: I am loading a shape-file from STATISTIK AUSTRIA (an offical authority in Austria) with the following projection:

PROJCS["MGI / Austria Lambert", GEOGCS["MGI", DATUM["Militar-Geographische Institut", SPHEROID["Bessel 1841", 6377397.155, 299.1528128, AUTHORITY["EPSG","7004"]], TOWGS84[601.705, 84.263, 485.227, 4.7354, -1.3145, -5.393, -2.3887], AUTHORITY["EPSG","6312"]], PRIMEM["Greenwich", 0.0, AUTHORITY["EPSG","8901"]], UNIT["degree", 0.017453292519943295], AXIS["Geodetic longitude", EAST], AXIS["Geodetic latitude", NORTH], AUTHORITY["EPSG","4312"]], PROJECTION["Lambert_Conformal_Conic_2SP", AUTHORITY["EPSG","9802"]], PARAMETER["central_meridian", 13.333333333333336], PARAMETER["latitude_of_origin", 47.5], PARAMETER["standard_parallel_1", 48.99999999999999], PARAMETER["false_easting", 400000.0], PARAMETER["false_northing", 400000.0], PARAMETER["scale_factor", 1.0], PARAMETER["standard_parallel_2", 46.0], UNIT["m", 1.0], AXIS["Easting", EAST], AXIS["Northing", NORTH], AUTHORITY["EPSG","31287"]]

Loading the shape-file seems to work fine: But when you make a pagelayout for a DIN A4 page you get this strange scale:

Setting the scale 1: 3 Million you get this:

I think QGIS uses weird/wrong Cartesian coordinates as my last screenshot suggests:

I installed QGIS with the standalone-installer on Windows 10.

Update: You can download the test-files (shp, prj) from this url: https://data.statistik.gv.at/data/OGDEXT_NUTS_1_STATISTIK_AUSTRIA_NUTS3_20190101.zip

Update #2: screenshot "about": Update #3: @gabriel-de-luca your workaround does not work for me: Cartesian coordinates seem to look okay now, but ellipsoid coordinates are completly wrong, and adding a scale-bar in the page-layout is completly wrong too:

Update #4: Screenshot Project Properties General:

Update #5: This weird behaviour does not occur under Ubuntu :

Update #6: A small screenshor recording of GQIS under Windows shows the strange behaviour. Look like the number of the kilometers becomes bigger, then smaller, than bigger again, .... Update #7: Pan and Zoom does not help, a basemap (NE 1: 50 Mio) is show as a basemap:

Answer 1

The definition of the CRS provided in the .prj file is a WKT1 string.

It seems that you have specified in the Settings that the project must have the CRS of the first layer added. And it seems that the project is not being defined with the EPSG:31287 CRS.

The Measurement tool takes the coordinates from the canvas.

For the ellipsoidal measurement, QGIS is required to project the canvas coordinates to the ellipsoid, that work is done by PROJ and it works well.

But for the Cartesian distance, QGIS only read the coordinates and compute the distance. QGIS is doing it, but doesn't know the units of measurement that is computing. (All units are returning the same distance.)

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A workaround is explicitly define the canvas CRS, trough Project ~> Properties ~> CRS menu or from the CRS icon, to EPSG:31287.
Note that you must define the Settings to Ask for datum transformation if several are available, because the default transformation between EPSG:31287 and EPSG:4326 is not the same as defined in the WKT1 string (MGI to WGS 84 (8) - EPSG:1194).

Another workaround is saving the layer to a GeoPackage. You will be asked to define the CRS and you can do it with the same WKT1 string (and make sure that QGIS will ask you about the preferred datum transformation).
GDAL will assign EPSG:31287 to the layer.
Start a new project. When the GeoPackage layer is loaded, QGIS will recognize its CRS and assign it to the canvas. In the project properties will be added the preferred datum transformation for that CRS.

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This issue was reported: https://github.com/qgis/QGIS/issues/34196.

This issue was fixed on February 3rd, 2020.