gdalinfo returns PROJCS["unnamed"] but QGIS returns the correct projection

Question

I'm trying to understand why my gdalinfo call to a specific file returns "unnamed" for the PROJCS node. The call looks like this: ./gdalinfo /path/to/file/dem.tif.

Output (notice the "unnamed"):

Driver: GTiff/GeoTIFF
Files: /path/to/file/dem.tif
Size is 25000, 25000
Coordinate System is:
PROJCS["unnamed",
    GEOGCS["WGS 84",
        DATUM["WGS_1984",
            SPHEROID["WGS 84",6378137,298.257223563,
                AUTHORITY["EPSG","7030"]],
            AUTHORITY["EPSG","6326"]],
        PRIMEM["Greenwich",0],
        UNIT["degree",0.0174532925199433],
        AUTHORITY["EPSG","4326"]],
    PROJECTION["Polar_Stereographic"],
    PARAMETER["latitude_of_origin",70],
    PARAMETER["central_meridian",-45],
    PARAMETER["scale_factor",1],
    PARAMETER["false_easting",0],
    PARAMETER["false_northing",0],
    UNIT["metre",1,
        AUTHORITY["EPSG","9001"]]]
Origin = (-1600000.000000000000000,2350000.000000000000000)
Pixel Size = (2.000000000000000,-2.000000000000000)
Metadata:
  AREA_OR_POINT=Area
Image Structure Metadata:
  COMPRESSION=LZW
  INTERLEAVE=BAND

I've tried to set both GDAL_DATA and PROJ_LIB environment variables but still, I don't get the right projection back.

Using QGIS 3.10 I get this:

PROJCS["WGS 84 / NSIDC Sea Ice Polar Stereographic North",
    GEOGCS["WGS 84",
        DATUM["WGS_1984",
            SPHEROID["WGS 84",6378137,298.257223563,
                AUTHORITY["EPSG","7030"]],
            AUTHORITY["EPSG","6326"]],
        PRIMEM["Greenwich",0,
            AUTHORITY["EPSG","8901"]],
        UNIT["degree",0.0174532925199433,
        AUTHORITY["EPSG","9122"]],
        AUTHORITY["EPSG","4326"]],
    PROJECTION["Polar_Stereographic"],
    PARAMETER["latitude_of_origin",70],
    PARAMETER["central_meridian",-45],
    PARAMETER["scale_factor",1],
    PARAMETER["false_easting",0],
    PARAMETER["false_northing",0],
    UNIT["metre",1,
        AUTHORITY["EPSG","9001"]],
AXIS["X",EAST],AXIS["Y",NORTH],AUTHORITY["EPSG","3413"]]

Why my gdal installation doesn't recognise the projection?

If I run ./gdalsrsinfo "EPSG:3413" the output is correct, similar to what QGIS is printing.

listgeo output:

TIFFReadDirectory: Warning, Unknown field with tag 42113 (0xa481) encountered.
Geotiff_Information:
   Version: 1
   Key_Revision: 1.0
   Tagged_Information:
      ModelTiepointTag (2,3):
         0                 0                 0
         -1600000          2350000           0
      ModelPixelScaleTag (1,3):
         2                 2                 0
      End_Of_Tags.
   Keyed_Information:
      GTModelTypeGeoKey (Short,1): ModelTypeProjected
      GTRasterTypeGeoKey (Short,1): RasterPixelIsArea
      GTCitationGeoKey (Ascii,8): "unnamed"
      GeographicTypeGeoKey (Short,1): GCS_WGS_84
      GeogCitationGeoKey (Ascii,7): "WGS 84"
      GeogAngularUnitsGeoKey (Short,1): Angular_Degree
      GeogSemiMajorAxisGeoKey (Double,1): 6378137
      GeogInvFlatteningGeoKey (Double,1): 298.257223563
      ProjectedCSTypeGeoKey (Short,1): User-Defined
      ProjectionGeoKey (Short,1): User-Defined
      ProjCoordTransGeoKey (Short,1): CT_PolarStereographic
      ProjLinearUnitsGeoKey (Short,1): Linear_Meter
      ProjNatOriginLatGeoKey (Double,1): 70
      ProjFalseEastingGeoKey (Double,1): 0
      ProjFalseNorthingGeoKey (Double,1): 0
      ProjScaleAtNatOriginGeoKey (Double,1): 1
      ProjStraightVertPoleLongGeoKey (Double,1): -45
      End_Of_Keys.
   End_Of_Geotiff.

Projection Method: CT_PolarStereographic
   ProjNatOriginLatGeoKey: 70.000000 ( 70d 0' 0.00"N)
   ProjStraightVertPoleLongGeoKey: -45.000000 ( 45d 0' 0.00"W)
   ProjScaleAtNatOriginGeoKey: 1.000000
   ProjFalseEastingGeoKey: 0.000000 m
   ProjFalseNorthingGeoKey: 0.000000 m
GCS: 4326/WGS 84
Datum: 6326/World Geodetic System 1984
Ellipsoid: 7030/WGS 84 (6378137.00,6356752.31)
Prime Meridian: 8901/Greenwich (0.000000/  0d 0' 0.00"E)
Projection Linear Units: 9001/metre (1.000000m)