Ou
grep -P -o 'Data.+?\d\d\d\d-\d\d-\d\d'
ou
perl -pe 's/^.+(Data.+?\d\d\d\d-\d\d-\d\d).+$//'
fará. Ambos imprimem a string mínima que começa com Data
e termina no que parece uma data (AAAA-MM-DD).
echo "fvvDataFolders/DDB/DDB2018-02-21oM]" > input.txt
echo "fbbDataFolders/DDB/DDB2018-02-22oM]" >> input.txt
grep -P -o 'Data.+?\d\d\d\d-\d\d-\d\d' input.txt
# output:
DataFolders/DDB/DDB2018-02-21
DataFolders/DDB/DDB2018-02-22
perl -pe 's/^.+(Data.+?\d\d\d\d-\d\d-\d\d).+$//' input.txt
# output:
DataFolders/DDB/DDB2018-02-21
DataFolders/DDB/DDB2018-02-22