"Normal" - hard to say, that's a relative term.
Permitted - yes. Here is a quote from the Microsoft SQL Server Express blog where LocalDB was introduced:
"...if the simplicity (and limitations) of LocalDB fit the needs of the target application environment, developers can continue using it in production, as LocalDB makes a pretty good embedded database too."
UPDATE (March 2019)
After using LocalDB in dozens of production sites for two solid years now, I'll confirm that it works. However, if you need a free multi-user ("shared" in LocalDB terminology) DB, use SQL Server Express instead. LocalDB is not a good multi-user solution. There are MANY obstacles to overcome. I don't have time now, but if you would like details on those obstacles and respective workarounds, leave a comment, and I'll document them here or in another question.
UPDATE (April 2021)
Showstopper -- the shared (i.e., multi-user) instance feature is broken in SQL Server LocalDB 2017 and 2019. Having one of those versions installed will even prevent accessing a shared instance created in an earlier version. The problem is documented here. The only known "workaround" is to uninstall the broken LocalDB versions, and use LocalDB 2016 or earlier.
UPDATE (September 2021)
There is now a fix for the Showstopper issue mentioned in the preceding update (UPDATE April 2021). I have successfully tested this fix for both versions. Minimum versions are: