The question is if a renter can significantly alter the property or the property's appearance - in this case the exterior with the signs. I doubt the fact that the alterations are signage is really significant; unless they are stolen, or they cause someone to make a wrong turn into the corner of the building. :)
It's probably in the lease that a renter can't make significant changes to the property's appearance without permission from the property owner or management; i.e. many leases don't allow you to repaint rooms, put in new carpet, etc. without permission from the landlord.
But, it all depends on the lease. If you both have the same property management company, see what is in your lease about property alterations.
Every lease I have ever seen (and most of the free template lease forms on the web have) had a clause stating that making unauthorized changes to the property is not permitted, along with the possible penalties, like paying for the damages, losing a deposit, even eviction.
The fact that you're in CA really doesn't matter when it comes to the lease; damage clauses, though pretty much universal, are by choice of the landlord. (As an aside, CA does have statewide laws about lead paint, for one example, that are required to be disclosed in leases.)
Is this legal?
This depends on the lease, as above.
What can the property management company and/or the property owner do
That's up to them, but a possibility is for someone to call the property manager and let them know what's happening.