I agree with you that it is possible to do it but it is a pretty big and challenging job that probably does not justify the effort to save a few hosting bucks given how much work would be involved in completing the migration.
While there is a correspondence between the tables there are differences in supported data types that would have to be mapped in queries created to migrate table data. For example MS SQL uniqueidentifer would map to a char/varchar column in MySql and bit columns would map to int or tinyint in MySql. And there are a lot of tables so it is a big project. If one really wants to do that work a tool like Aqua Data Studio is probably what I would use.
Hope that helps,