Like Ciaran I'm not very familiar with GDAL's python bindings, but I've tried running a little snippet which I believe does the same thing as what was done with rasterio: it writes a JP2 to a vsimem in-memory file, and then writes the bytes to a file on disk.

from osgeo import gdal

mem, out = "/vsimem/B01.jp2", "new_B01.jp2"
gdal.Translate(mem, "B01.jp2")

# Taken from https://lists.osgeo.org/pipermail/gdal-dev/2016-August/045030.html
f = gdal.VSIFOpenL(mem, "rb")
gdal.VSIFSeekL(f, 0, 2)  # seek to end
size = gdal.VSIFTellL(f)
gdal.VSIFSeekL(f, 0, 0)  # seek to beginning
data = gdal.VSIFReadL(1, size, f)
gdal.Unlink(mem)

with open(out, "wb") as ff:
    ff.write(data)

When I look at the gdalinfo of new_B01.jp2, it has a CRS and a transform, so I am not able to reproduce the behavior we have with rasterio.

(Side note: Interestingly enough, the pixel values of new_B01.jp2 have slightly changed with respect to the original B01.jp2 file, so there is some data loss somewhere in the process. But maybe that is expected of the JP2 format and could be avoided by passing extra arguments to gdal.Translate? I mainly have experience with writing GeoTIFFs, writing JP2s is new to me)

Guillaume