Live website while Maintaince

You can’t. However, updates usually only take a few seconds, which is why this should not be such a big problem.

You can also control whether updates happen automatically or only manually. You can find some information about this here: https://wordpress.org/documentation/article/configuring-automatic-background-updates/ – there are also plugins that simplify the configuration if you don’t want to handle source codes yourself.

Or is it the case that the maintenance status is not terminated even after the updates have been completed? This indicates an error in the updates. They were obviously not completely finished. Details should be visible in the error log of your hosting.