I would recommend you use two different databases for that. Otherwise, you will end up mixing data for both customers. You really want client 1 to change data, then logout, login as client 2 and suddenly all pending client 1 not belongs to client 2? It doesn't look it's desired behavior.
But you're right, it's better to use multiple databases (one for each customer).
Considering this scenario, on my server: Can I have just one database (containing the records of all clients) or should I also have a database for each client?
Yes, you can have one single database in the server. Actually each client database is a different node, it doesn't matter if it's in the same computer or a different one.
Suppose I already have a database with records (managed by Aurelius/Echo). If I want to create a new client, is it my responsibility to insert the records that are already in the server's database into the new client? Or is echo able to sync them?