MySQL and PostgreSQL Licence issue Our Company wants to use MySQL for one software product. This product is not open-source. Is it legal to use it freely ?

If not, how about PostgreSQL ? Is this okay for our purpose ?
Postgres (not Postgre) is absolutely free. You can do anything you want with it. 

For MySQL you will most probably need a commercial license if you plan to redistribute your application with a bundled database. Not sure if this is required if your application just needs MySQL but I think this also requires a commercial licenses because it is considered a derivative work

But I'm not a lawyer. So if you want to be 100% sure, talk to Oracle and a lawyer.