Modbus TCP together with MQTT over same ethernet
Does it work if I simply add a network switch and connect the sensor and an internet router to it, and use both protocols in parallel?

Kind of, yes. You just connect both the MODBUS device and the Arduino to the network provided by the router. The Arduino is just a normal network device - the IP address for it allocated by the router, as is the MODBUS device.
Any device can communicate with any other device in a network.