Sample code:
Component:

<aura:component implements="lightning:actionOverride, lightning:isUrlAddressable">
    <aura:attribute name="recordId" type="Id"/>
    <aura:handler name="init" value="{!this}" action="{!c.doInit}"/>
    Record Id is {!v.recordId}.
</aura:component>

Controller:

({
    
    doInit : function( component, event, helper ) {
        
        let pageRef = component.get("v.pageReference");
        console.log( 'Page Ref is ' + JSON.stringify( pageRef ) );
        let state = pageRef.state; // state holds any query params
        let base64Context = state.inContextOfRef;
        if ( base64Context.startsWith("1\.") ) {
            base64Context = base64Context.substring( 2 );
        }
        let addressableContext = JSON.parse( window.atob( base64Context ) );
        component.set( "v.recordId", addressableContext.attributes.recordId );
        
    }
    
})

New Button Override:

Output: