Need help with using session.setItem and session.getItem

Regarding your code, everything looks just well.
Testing your code —> also works like a charm!

Just a simple testing page…

https://russian-dima.wixsite.com/my-site-4/blank-6

Page-1 (sendind…)

import wixlocation from 'wix-location'
import {session} from 'wix-storage';


$w.onReady(()=> {   
    $w('#button1').onClick((event) => {    
        session.setItem("stage", "1");
        console.log('Stage selected is ' + session.getItem("stage") );
        wixlocation.to("https://russian-dima.wixsite.com/my-site-4/reciever-site");
    });   

    $w('#button2').onClick((event) => {
        session.setItem("stage", "2");
        console.log('Stage selected is ' + session.getItem("stage") );
        wixlocation.to("https://russian-dima.wixsite.com/my-site-4/reciever-site");
    });   

    $w('#button3').onClick((event) => {
        session.setItem("stage", "3");
        console.log('Stage selected is ' + session.getItem("stage") );
        wixlocation.to("https://russian-dima.wixsite.com/my-site-4/reciever-site");
    });   

    $w('#button4').onClick((event) => {
        session.setItem("stage", "4");
        console.log('Stage selected is ' + session.getItem("stage") );
        wixlocation.to("https://russian-dima.wixsite.com/my-site-4/reciever-site");
    }); 

    $w('#button5').onClick((event) => {
        session.setItem("stage", "5");
        console.log('Stage selected is ' + session.getItem("stage") );
        wixlocation.to("https://russian-dima.wixsite.com/my-site-4/reciever-site");
    });  
});

Page-2…(receiving):

import {session} from 'wix-storage';

$w.onReady(()=> {    
    session.getItem("stage");
    console.log('Stage received is ' + session.getItem("stage"));
});

Do the test and take a look at console.