No, You cannot.
Shared plans are designed to fulfill the needs of customers having different needs, so type of database is an open option for them. They use what they want to.
You can have another MySql/MSSql DB for just $2.00/db/mon. I think it is reasonable to pay extra for your specific needs.
However, Resellers are free to offer one or more of any or all DB(s) to their clients.
Well, seeing as mysql is a free db & your service is configurable I thought in the name of customer service this would just be a mater if adding another myqsl db & removing postgre in your control panel. Why should I pay an extra $2. a month for something that would take 5 minutes to set up & not cost jodohost anything. Or am i wrong is mqsql not an open source db?
MySql is free for personal use, but for commercial use there is a licensing fee. Because jodohost is providing a commercial service (regardless of how you plan on using the mysql), they need a license. https://order.mysql.com/ .
MySQL is free for commercial use as well, it is just dual licensed so that companies that want to develop with it don't have to opensource their apps because of using mysql libs/clients in their product if they include it.
Just as a clarification, their commercial license doesn't mean any commercial operation needs to have a commercial license.