And for the Driver() object?

The Driver() is limited compared to SB(). Either use SB() if you need the extra methods/functionality, or create your own methods that do what you need.

Hello, Michael. Why does local storage variable, which i've set via set_local_storage_item, after refreshing page is deleted?

 with SB(
 uc=True,
 browser="chrome",
 locale="ru",
 position="0,0",
 agent=browser_config.get("user_agent"),
 window_size=browser_config.get("window_size"),
 ad_block=True,
 devtools=True
 ) as driver:

 def login(driver: SeleniumBase, token: str):
 logger.info("Execute token to LocalStorage")
 temp = json.dumps({"token": token})
 driver.set_local_storage_item("wbx__tokenData", temp)

While we were working with raw selenium we did it like this:

 driver.execute_script("localStorage.setItem(arguments[0], JSON.stringify({token: arguments[1]}));",
 'wbx__tokenData', token)

And it were working fine.

As of 4.35.7 you have these CDP Mode methods:

sb.cdp.set_local_storage_item(key, value)
sb.cdp.get_local_storage_item(key)

And if you're calling things from regular mode, use:

sb.set_local_storage_item(key, value)
sb.get_local_storage_item(key)

Any method called from driver is using the raw Selenium API... (not the SeleniumBase one).
If there's an issue with a raw Selenium method, then you would have to check with their repo.