You are looking for a template with a self reference filter and that does exactly what you are looking for.
When you are in B database and click a new item, it will automatically create that entry for only B and you will only see B items.
For example, you can have a tasks database and a clients database. Inside each client you can add a view of the tasks database and filter it by the client itself, so you only see tasks for that client.
A template with that tasks view can be created and it can be filtered as self reference, so every time you create a new client, when creating the tasks view (from then template), it will automatically be self filtered to that new client.
Not easy to describe a step by step, but give it a try and also a quick search in YouTube can help.