MSPM0G1507: Are there any registers or settings that can recognise a state where nothing is connected to the input channel of the ADC (open state)?

You might get some mileage out of the internal pull-up/-down. This wouldn't be a definitive test, but it would tell you whether the impedance is >Rpullup (~40K), where "open" would be much larger. If the expected source impedance is small enough that could be something.

It will help, while it will do imapct on the ADC conversion data, so user should evalute the impact on the ADC conversion.

That's kind of the idea -- if the (weak-ish) internal pull-up/-down can influence the ADC reading, then the pin's input impedance must be much higher than the internal resistor, which could mean "disconnected". It sets up a resistor divider where the low/high side is unknown but probably (over a short time) roughly constant.

Such a probe would be pretty quick, and generally benign.

It wouldn't tell you much if the sensor (ADC input) were expected to be very high impedance.

Yes, depends on the requirements.