BDO is two data centers;
one for NA,
one for EU. If you make a family on one, when you switch over, it'll prompt you to create a new family name, meaning it's not interchangeable as your statement would lend one to believe. The other "servers" in BDO would be called channels/realms/shards in some other games, like TERA. Adding a new channel/realm/shard isn't nearly as costly as adding a new server or data center, but is meant to split the player-base apart to avoid things like overloading the server. BDO's servers are all of the regions (Olvia, Velia, Balenos, Serendia, Mediah, Calpheon, Kamasylvia, Arsha) - they share certain chat features when you are within that server. The shards are numbered (Calpheon 1 is the first shard of the server Calpheon; Calpheon 1 to 6 will share the same server chat, but when you switch to Mediah, you stop hearing Calphone's chat and gain access to Mediah 1 to 6's instead). World chat broadcasts over the entirety of the data center.
FFXIV is devoid of the shard-server model; the server
is the channel/realm/shard.
Links are what I'm greeted with when I pick between NA and EU DCs in BDO.