There are many kinds of databases. Graph, key/value, relational, hierarchal etc. That doesn't change the fact that any app that writes code to store and retrieve data is creating their own database or using someone else's.

There are many kinds of data models. A DBMS is a DBMS, and typically a specific DBMS supports a specific data model. A log structured file is at most a data store.

To be fair this is a somewhat fuzzy categorization (DBMS, DB, Data Store) and it can cause confusion. An DBMS is a system, just like it says right on the acronym tin. A DBMS is a system that provides capabilities such DDL, DML, auxiliary processes, etc. to manage a database. There are various data models, e.g. a triple store, for databases, but the data model X is a orthogonal to XDBMS.