There two aproaches:

Use project dependency

Form Playwright 1.31 with project dependency you can set test what will be executed before any other tests and pass i.e. browser storage between tests.

See working and tested example (entering google search page and accepting policies, and then entering this page again with cookies):

playwright.config.ts:

import { PlaywrightTestConfig } from "@playwright/test";
import path from "path";

//here you save session
export const STORAGE_STATE = path.join(__dirname, 'login.json')
    
const config: PlaywrightTestConfig = {
  timeout: 10 * 1000,
  expect: {
    timeout: 3 * 1000,
  },
  testDir: './tests',
  use:{
    baseURL: 'https://google.com',
    trace: 'retain-on-failure',
    video: 'retain-on-failure',
  },
  // here we set main project and dependent one
  projects: [
    {
      name: 'login',
      grep: /@login/
    },
    {
      name: 'depend e2e',
      grep: /@e2e/,
      dependencies: ['login'],
      use: {
        storageState: STORAGE_STATE
      }
    }

  ]
}; 

export default config;

tests/example.spec.ts:

import { test} from '@playwright/test';
import { STORAGE_STATE } from '../playwright.config';
    
test('login to service @login', async({page}) => {
  await page.goto('/');
  // below example is to reject cookies from google
  await page.waitForLoadState('domcontentloaded');
  await page.keyboard.press('Tab');
  await page.keyboard.press('Enter');
  await page.getByRole('menuitem', { name: "English (United States)" }).press('Enter');
  await page.getByRole('button', { name: 'Reject all' }).click();
  
  // Save storage:
  await page.context().storageState({path: STORAGE_STATE})
})

test('logged in test @e2e', async ({page}) => {
  await page.goto('/');
  await page.waitForLoadState('domcontentloaded');
  // BOOM - you are in!
  // Screenshot shows that settings regarding cookies were saved
  await page.screenshot({ path: 'screenshot.png' })
})

Video with example: https://www.youtube.com/watch?v=PI50YAPTAs4&t=286s

In above example I'm using tags to identify tests: https://playwright.dev/docs/test-annotations#tag-tests

Use fixtures

Let test entering google search page and accepting policies

In file fixtures.ts:

import { test as base } from "@playwright/test";

export const test = base.extend({
  page: async ({ baseURL, page }, use) => {
    // We have a few cases where we need our app to know it's running in Playwright.
    // This is inspired by Cypress that auto-injects window.Cypress.
    await page.addInitScript(() => {
      (window as any).Playwright = true;
    });

   await page.goto("/");
   // below example is to reject cookies from google
   await page.waitForLoadState("domcontentloaded");
   await page.keyboard.press("Tab");
   await page.keyboard.press("Enter");
   await page
     .getByRole("menuitem", { name: "English (United States)" })
     .press("Enter");
   await page.getByRole("button", { name: "Reject all" }).click();

    use(page);
  },
});
export { expect } from "@playwright/test";

And then in test use new test object instead importing it form "@playwright/test"

import { test, expect } from "../fixture";

test("logged in test @e2e", async ({ page }) => {
  await page.goto("/");
  await page.waitForLoadState("domcontentloaded");
  // BOOM - you are in!
  // Screenshot shows that it works
  await page.screenshot({ path: "screenshot.png" });
});

Inspired by: https://github.com/microsoft/playwright/issues/9468#issuecomment-1403214587

If you need example with global request intercepting see here: https://stackoverflow.com/a/76234592/1266040

The solution is very simple. It is executing as an independent test since you are passing {page} in each test, so if you want to use the same context for both the test you need to modify the test like below.

import { test, expect } from '@playwright/test';

test.describe('Add a simple invoice test', () => {
  let page: Page;
  test.beforeAll(async ({ browser }) => {
    page = await browser.newPage();
  });

    test('01.Login & add an invoice', async () => { // do not pass page
        await page.goto("https://someUrl.com");
        await page.fill('input[id="email"]', "someEmailAddress");
        await page.fill('input[ng-model="ctrl.user.password"]', "somePassword");
        await page.click('button[id="login-btn"]');
    });

    test('02.Add an invoice', async () => { //do not pass page 
        await page.click('[name="invoice"]');
        await page.click('button[id="addInvoiceButton"]');
        await page.click('a[ng-click="ctrl.goToAddInvoice()"]');
        await page.fill('#invoiceTitle', Math.random().toString(36).substring(7));
        await page.fill('#exampleInputAmount', "120");
        await page.click("#invoiceCategory")
        await page.fill("#invoiceCategory > input", "Car")
        await page.keyboard.press("Enter");
        await page.click('button[id="submitInvoiceButton"]');
    });
});

This should work as you expected You can also refer to the Dzone Article regarding the same.

Note: Playwright doesn't recommend this approach but this answer should fulfill your need.