There’s a new bug with Access 2016 in which the Open last used database when Access starts property does not work.
This issue also seem to involve hung msaccess.exe process, so Access does not appear to be shutting down properly. When exiting Access, be sure to use check the Windows Task Manager for hung processes and end them.
A Few Threads on the Subject
Office 2016 Access not automatically opening last database used.
ACCESS 2016
Build 1705 and Client Settings
MS Access 2016 will not release CPU after shutting down.
The Confirm Action Queries, Document Deletions and Record Changes options will not stay unchecked.
Update
The bug in questions appears to be much more generalized and impacts many, if not all, of the database options/settings. So the workarounds below can be generalized, with the obvious need to first identify each options corresponding registry key (most are pretty obvious).
Workaround No.1
I did a quick test and isolated the RegKey, if you want to manually set/reset it, I believe you should be able to by editing
HKEY_CURRENT_USER\Software\Microsoft\Office\16.0\Access\Settings\
And then edit the LastLoad DWord with the following hexadecimal value
0 = Do not load the last used database automatically
1= Load the last used database automatically
The issue has been confirmed by multiple sources and I have already advised the Microsoft Access Development Team of the issue. All we can do now is wait for them to push out a patch/fix.
Workaround No2.
Today, May 9th, I was offered another workaround from the Access Dev Team that is looking into the issue. Simply open a form (any form!) in Design view prior to opening the Database Options. It does appear to work based on my brief testing and at least this way you don’t need to mess around with the registry directly.
Possible Side Effect With RibbonCreator
Fellow MVP George Hepworth has flagged the fact that it would appear that this issue seems to impact the RibbonCreator and it generates the following error
Do note that a proposed workaround can be found at Exporting to an Access database ends with the Message “The selected database is open. …” which basically states to perform an Office Repair.
Do also note that after performing a 2016 Office Repair, I lost all my taskbar links, and Office was deactivated and then would not recognize my account and I had to contact MS support to get it operational again!
Update From The Access Dev Team
The Access Dev Team shared with me the fact that:
A fix is on its way in the May fork, which should have build numbers like 16.0.8229.*, and be available to Current Channel customers in early July. You can share that information (although dates for future releases are always estimates).
So hopefully this will be resolved in the near future.

Suggest that the Team compare the code between the Exclusive and Shared modes. There is no problem with Access 2016 opening the previously opened file when it runs in the Exclusive mode. The solution could be as simple as that. Hopefully it is.
Thank you for sharing your observations.
If your suggestions is that by opening a database file in Exclusive mode makes these properties stick, my tests show this not be the case. Could you elaborate further what you meant?
My issue is similar (related?) but not identical. My Access loads only one db over and over upon start, regardless of the last one used. I have turned on and off “load last” numerous time but anytime it’s turned “on” the same db always loads. And it’s an obscure one I haven’t used for months. Guess I’ll just leave it “off”for now.
Quite interesting!
Thanks