This is adaptor/base for button assembly made by Aptimex.  (https://makerworld.com/en/models/628744-18mm-button-assembly-for-6mm-push-buttons#profileId-553389) This adaptor supports button so that pins wont bend when button is pressed. There is also 3 button base for easier use.

By soldering dupont pins to buttons legs, you can add dupont wires straight to the button and use these separately from your breadboard. You can also simply solder wires themselves to button legs. (See examples from bad pictures, I dont have other colors atm)
After you have soldered connectors/wires, you can simply drop the button assembly into base.