There is alway an "it depends" aspect to these type of questions, however let's give it a try. I will give 3 scenario's which you have determine which best describes your situation.
Proof of concept / v1
If you just create a simple chatbot, without expectation of large traffic you might as well use any DBMS of choice. SQLite might be perfect for a proof of concept. No need to make it perfect on your first try.
For a regular user group
You can consider one of the many general use databases. SQL or NoSQL, whatever fits. I personally prefer SQL because you can more easily force data integrity and relationships between data, but some people prefer NoSQL because of the freedom it provides. There are many articles that discuss both the pros and cons of both camps. I will not expand on that here, please just take the time to seek out your own preference. As for SQL I would recommend looking into MariaDB or Postgres and for NoSQL just Mongo.
For large scale chat
If you are expecting to have a very very big group of users that will talk to your chatbot all the time, it might be a good idea to "bring out the big guns". Why don't we just borrow the techniques used by well known big companies that provide chat, like Facebook using HBase and Discord using Cassandra. Which interestingly are both Apache projects.
You might want to read up on ORM's where you can switch the underlying Database on the fly when supported.