I really like the abstract database API that accompany Django, I'm wondering basically can use this (or something like that similar) to model, access, and manage my (postgres) database for my non-Django Python projects.

What you are searching for is definitely an object-relational mapper (ORM). Django features its own, built-in.

To make use of Django's ORM alone:

If you wish to use another thing:

Popular stand-alone ORMs for Python:

All of them support MySQL and PostgreSQL (amongst others).

I especially like SQLAlchemy with following tools:

They help remind me of ActiveRecord.