I can think of two possible reasons (off the top of my head) you may be seeing this error:
1) The two Server licenses are attempting to use the same key. Since they're on different networks, they don't "see" each other until you attempt to connect to both.
2) You have an abnormal termination of your own FM client. This can happen (especially in a Windows environment), and when it does, Server still sees the process running, thinks there's more than one client using the same license, and forces you off. Solution: Shut down your computer and try again.
If neither of those works, post back and maybe we can come up with something else with more detail.
I ran into this EXACT thing with a legacy multifile system where there was more than one FileMaker Data Reference in a file to another file.
Check your external data references in each file, and repoint any relationship graph tables to use a single file reference in the data file list for one file.
If you access a table which uses a second reference to the same file via a different data source entry, the server reads it as your License trying to open the file more than once at the same time, and the livence conflict kicks in.
This didn't happen in FM10, but it started becoming more picky with 11 (when it first bit me), and is still an issue in 12 if you don't keep the FileMaker Data Source list very clean.
It is impossible to know exactly which set of file references is causing the error without know the definition of every file reference. And every reference used in every file opened by a file reference or a script. You state you are working in two 'databases' (files) but do they in turn open other files?
If you set up file references using an IP address on one and a DNS name inn another, maybe even in two different files, then FMS may see them as two different connection requests if it comes through two different routers and therefore it must be from two different computers. This primarily happens when the IP's of one LAN do not map or are duplicated in a second LAN over the WAN. Or FMP cannot find the original IP, in this case perhaps the new server did not get assigned the same IP address, but a DNS exists for the name used in or by a different File Reference. Without know exactly what all the file references are between the two files and what other files are used in the systems, and we also have their file references to consider.
The only suggestion I can make is look at every file reference in every file in the solution for mismatched type of reference (IP vs DNS). I suggest making all of them DNS names. I try not to use IP addresses as they can change and you must then update every File reference in every file connecting to that file to use the new IP.
I experienced the same problem when opening 2 different sets of files on the same server, but using different IP addresses in the file reference. For example, one of the files accessed another file (on the same server) via a file reference using a LAN address (192.168... etc), while a different file accessed the server-hosted file using a different IP address (25... etc). It will hopefully help to check all the file references and make them consistent as indicated by ch0c0halic.
Thanks everyone for the good information. I'll check on each of the suggestions.