Does this sort of "multi lane" switch exist?

Each of your "lanes" is a "pole" in switch terminology. 1P (or SP), 2P (or DP), 3P, 4P...

The number of ways you can connect those to outputs is a "throw" (T) - Single being ST, double being DT and after that, numbers. 3T, 4T...

Some switches add "center-off." Single-throw switches are "off" one way and "on" the other, while DT are always on in one direction or the other UNLESS they are "center off."

3P DT will do. Input goes to common.

I have some lovely old modem A/B switches that switch all 25 pins (though most do fewer pins, all 25 being overkill for the average modem) so those are 25P-DT

NKK M-2033 3PDT switch - This 3PDT switch may meet your needs. Se datasheet for connection and ratings information.

This is the relevant pole & circuit table from the datasheet: