The official Oracle MySQL connectors seem to correspond with these GitHub repositories:
They all contain a LICENSE.txt that specifies that the connector is released under version 2 of the
GNU General Public License (GPLv2), but with the following additional permissions:
This distribution of MySQL Connector/C++ 8.0 is distributed with certain software (including but not limited to OpenSSL) that is licensed under separate terms, as designated in a particular file or component or in the license documentation. Without limiting your rights under the GPLv2, the authors of MySQL hereby grant you an additional permission to link the program and your derivative works with the separately licensed software that they have included with the program.
Without limiting the foregoing grant of rights under the GPLv2 and additional permission as to separately licensed software, this Connector is also subject to the Universal FOSS Exception, version 1.0, a copy of which is reproduced below and can also be found along with its FAQ at http://oss.oracle.com/licenses/universal-foss-exception.
Considering the scenario of a backend of commercial closed-source website. Would the backend be allowed to use these connectors to connect to a MySQL instance, or would the usage of the connectors require to open source the entire web application backend code?
stackexchange.com. For now the backend is hosted fully closed. It is only "distributed" in the sense that it is deployed to backend node instances -- not sure if this can fall under distribution depending on how a cloud hosting provider implements it. But who knows, one day the backend software may have to be distributed publicly (e.g. to allow other self-hosting) or even open sourced.