How does AuthzSVNAccessFile work?

Does the error only occur for user doe on both repositories? Or does it fail for grig on the second repository? Assuming it fails for all users, and assuming the error isn't in the AuthUserFile and the SVNParentPath is correct, I think you need to add a default access rule for all of your repositories.

[/]
* = r

[first:/]
doe = rw

[second:/]
doe = rw
grig = rw

Or you could put users doe and grig into a group and do it as follows:

[groups]
secondteam = doe, grig

[/]
* = r

[first:/]
doe = rw

[second:/]
@secondteam = rw

You mention that you would like to better understand how the AuthzSVNAccessFile is supposed to work. I recommend reading this tutorial. A full path-based authorization file example from that tutorial for supporting multiple repositories is below:

[groups]
admin = john, kate
devteam1 = john, rachel, sally
devteam2 = kate, peter, mark
docs = bob, jane, mike
training = zak

# Default access rule for ALL repositories
# Everyone can read, admins can write, Dan German is excluded.
[/]
* = r
@admin = rw
dangerman =

# Allow developers complete access to their project repos
[proj1:/]
@devteam1 = rw
[proj2:/]
@devteam2 = rw
[bigproj:/]
@devteam1 = rw
@devteam2 = rw
trevor = rw

# Give the doc people write access to all the docs folders
[/trunk/doc]
@docs = rw

# Give trainees write access in the training repository only
[TrainingRepos:/]
@training = rw

did you use "htpasswd" to add doe user to /home/svn/dav_svn.passwd or it was from some type of front-end admin ?

i had problems for example with Virtualmin/Webmin they were editing the htpasswd file in a format diffrent from apache, and the problem was solved by readding the user using htpasswd.