The fact that you can see the error message on your terminal means it's not going into the log file.
As others have mentioned, you are redirecting stderr from the gzip element of the pipeline, not the mysqldump element.
If you think about your pipline as a series of elements, it may become clear what's going on:
mysqldump -u root -p$Pass --all-databases
gzip > fulldbdmp-$(date +%m%d%Y).dump.gz 2> file.log
You were probably after this:
mysqldump -u root -p$Pass --all-databases 2> file.log
gzip > fulldbdmp-$(date +%m%d%Y).dump.gz
Which becomes:
mysqldump -u root -p$Pass --all-databases 2> file.log \ |
gzip > fulldbdmp-$(date +%m%d%Y).dump.gz
Side note: I would strongly encourage you not to ignore the following warning. It is there for a reason. see 6.1.2.1 End-User Guidelines for Password Security
Warning: Using a password on the command line interface can be insecure