My guess (never worked with DAT) is what they have them in audio format, not DDS, which is digital but...
Judging on some cursory search[0] the only way to transfer the audio is to hook the drive through S/PDIF and play the whole tape. Looks like the ability to treat audio as the data (similar to the way we can treat AudioCDs as just a data disk filled with PCM) is rare and not quite the official [1]
Anyway, you now need to acquire such drive, get to @jesprenj institution, tell them they are doing it wrong, threw out the current setup, replace the drive and the 'recording' station....
With a 30 y.o. tech which is obsolete for 20 years it's all.. questionable.
Open Source software exists[0] for the purpose.
0. https://github.com/andrew-taylor/read_dat