Django apps don't segregate objects, they are a way of structuring your code base. The idea is that an app can be re-used in other projects. Having a separate app for your catalogue of items and your projects is a good idea, but having them together in one is not a problem if you have a small codebase.
If I have understood your post correctly, what you want is for the databases of different departments to be separate. This is essentially a multi-tenancy question which is a big topic in itself, there are a few options:

Code separation - all of your projects/departments exist in a single database and schema but are separate by code that filters departments depending on who the end user is (literally by using Django .filters()). This is easy to do but there is a risk that data could be leaked to the wrong user if you get your code wrong. I would recommend this one for your use-case.

Schema separation - you are still using a single database but each department has its own schema. You would need to use Postgresql for this but once a schema has been set, there is far less chance that data is going to be visible to the wrong user. There are some Django libraries such as django-tenants that can do a lot of the heavy lifting.

Database separation - each department has their own database. There is even less of a chance that data will be leaked but you have to manage multi-databases and it is more difficult to scale. You can manage this through django as there is support for multi-databases.

Application separation - each department not only has their own database but their own application instance. The separation is absolute but again you need to manage multiple applications on a host like Heroku, which is even less scalable.
I am currently building a tool in Django for managing the design information within an engineering department. The idea is to have a common catalogue of items accessible to all projects. However, the projects would be restricted based on user groups.
For each project, you can import items from the catalogue and change them within the project. There is a requirement that each project must be linked to a different database.
I am not entirely sure how to approach this problem. From what I read, the solution I came up with is to have multiple django apps. One represents the common catalogue of items (linked to its own database) and then an app for each project(which can write and read from its own database but it can additionally read also from the common items catalogue database). In this way, I can restrict what user can access what database/project. However, the problem with this solution is that it is not DRY. All projects look the same: same models, same forms, same templates. They are just linked to different database and I do not know how to do this in a smart way (without copy-pasting entire files cause I think managing this would be a pain).
I was thinking that this could be avoided by changing the database label when doing queries (employing the using attribute) depending on the group of the authenticated user. The problem with this is that an user can have access to multiple projects. So, I am again at a loss.
Handling repetitive content within django apps