Analysis modes are locked in Constraints Manager

It sounds like there is a restriction on the board file. You can lock the design with a password which may restrict Constraint changes. Look at File - Properties - Locking tab and see if there is password control on the board. You would need to get that password to unlock it. 

Unfortunately this is not the case ;-(
The design is not locked. Moreover I tried with different designs and in the Allegro PCB Editor 17.4 and 23.1 - both with hotfix installed.

Do you mean Environment Variables? I am using Windows 10.

There is no such variables in my system.
I also check the system and the user (account) variables.

OK Can you close the tools, open a Windows File Explorer and type %HOME% followed by a return into the address bar followed by a return. This opens your HOME folder (usually C:\SPB_Data or simlar). In here rename the pcbenv folder to pcbenv_old. Then restart the tools and try again. If this works you may need to redefine your library paths and other shortcut keys you have in the env file, if it doesn't delete the new pcbenv folder created and rename the pcbenv_old back to pcbenv.

I also made one change to Defined Variables and it's seen under Tools-Utilities-Env Variables
"set retain_electrical_constraints_on_nets = NO"
and no result :-(
I looked at all my *.cpm files for "set retain_electrical_constraints_on_nets = YES" but no hits found.

I deleted the pcbenv file - no positive result.
I also ran Allegro in safe mode - no positive result.

Can it be a setting in registry?

I don't think there is a registry setting for this but you maybe have to ask Cadence via the support site. What happens if you try a new blank board (File - New) then before you do anything try the constraint modes. If that's OK are the constraints controlled in the front end (DEHDL or System Capture). Do you have a site.cpm set? Is there anything there? Are there other users with the same issue?

I noticed a thing in allegro.jrl file which records all you do in the project.

\i (00:20:32) QtSignal m_dockWsQtWidgetContents m_domainSelector currentChanged Manufacturing
\i (00:20:32) QtSignal m_domainSelector WS_Manufacturing itemSelectionChanged "Design for Assembly" +
\i (00:20:32) QtSignal "DFA Constraint Set" Outline
\i (00:20:32) QtSignal Frozen_All_Layers_WorksheetView174 Frozen_All_Layers_WorksheetView174Header sectionMoved +
\i (00:20:32) QtSignal 0 0 1
\i (00:20:32) QtSignal Frozen_All_Layers_WorksheetView174 Frozen_All_Layers_WorksheetView174Header sectionMoved +
\i (00:20:32) QtSignal 1 1 0
\i (00:20:32) QtSignal All_Layers_WorksheetView174 All_Layers_WorksheetView174Header sectionMoved +
\i (00:20:32) QtSignal 0 0 1
\i (00:20:32) QtSignal All_Layers_WorksheetView174 All_Layers_WorksheetView174Header sectionMoved +

Many times the Frozen is there whenever I tried to change the status in Checkboxes. Maybe this is the direction where we should look for solution?

I look inside the *.cpm file there is nothing weird there.
I also ran "dbdoctor" check the database. Nothing helped.

I noticed a thing in allegro.jrl file which records all you do in the project.

\i (00:20:32) QtSignal m_dockWsQtWidgetContents m_domainSelector currentChanged Manufacturing
\i (00:20:32) QtSignal m_domainSelector WS_Manufacturing itemSelectionChanged "Design for Assembly" +
\i (00:20:32) QtSignal "DFA Constraint Set" Outline
\i (00:20:32) QtSignal Frozen_All_Layers_WorksheetView174 Frozen_All_Layers_WorksheetView174Header sectionMoved +
\i (00:20:32) QtSignal 0 0 1
\i (00:20:32) QtSignal Frozen_All_Layers_WorksheetView174 Frozen_All_Layers_WorksheetView174Header sectionMoved +
\i (00:20:32) QtSignal 1 1 0
\i (00:20:32) QtSignal All_Layers_WorksheetView174 All_Layers_WorksheetView174Header sectionMoved +
\i (00:20:32) QtSignal 0 0 1
\i (00:20:32) QtSignal All_Layers_WorksheetView174 All_Layers_WorksheetView174Header sectionMoved +

Many times the Frozen is there whenever I tried to change the status in Checkboxes. Maybe this is the direction where we should look for solution?

I look inside the *.cpm file there is nothing weird there.
I also ran "dbdoctor" check the database. Nothing helped.