It appears that I have clicked that update button little to fast... again...
If I have read official Release News The Joomla! 3.6.1 Update
a little bit more carefully, I would have learned following from the third paragraph:
Updating from Joomla! 3.6.0
Update the Joomla! Update Component through the Extension Manager. Then use the Joomla! Update component to update to the latest 3.6.x version.
So I have tested this statement on second test site, and sure enough: After I clicked on extensions Updates are available! quickicon in control panel, among other updates there was that important one: Joomla! Update Component Update 3.6.1
I have updated that one first, and then went back to Joomla Update component and clicked on that blue button 'Install the update', this time without any problems or error messages.
Later on I have seen few tweets from Joomla PLT members, referring to official Joomla Github Release Joomla 3.6.1
which helps a little bit further:
IMPORTANT! Due to a security update in this release please upgrade to Joomla 3.6.0 first then install this new version of the Joomla Update Component through Extension Manager before continuing upgrading Joomla as normal**
There are also download links to all important packages in questions, even to that Joomla Update Component update itself com_joomlaupdate-3.6.1.zip, just in case that for some reason that update doesn't appear in your Extensions Update component on your site.
Moral of the story: RTFM!
The good folks from JoomlaShack (ex-Alledia) just published What to Know About Updating to Joomla 3.6.2