Unable to save database diagrams

Once you resolve the ability to save database diagrams, please note that in the next release of SSMS v18 Microsoft is deprecating database diagram feature. I don't quite understand this thinking however you can find more information in the release notes.

- October 3, 2018

Rest of the Story:

It sounds like your sysdiagrams table somehow lost the IDENTITY property on the diagram_id column.  Using the following SQL I was able to drop the sysdiagrams table and rebuild.  Note you will lose any existing diagrams (if you had any during this process)

DROP TABLE dbo.sysdiagrams;
CREATE TABLE [dbo].[sysdiagrams]
    [name] [sysname] NOT NULL,
    [principal_id] [int] NOT NULL,
    [diagram_id] [int] IDENTITY(1,1) PRIMARY KEY,
    [version] [int] NULL,
    [definition] [varbinary](max) NULL,
    CONSTRAINT [UK_principal_name] UNIQUE ([principal_id],[name])

GO EXEC sys.sp_addextendedproperty @name=N'microsoft_database_tools_support', @value=1 , @level0type=N'SCHEMA', @level0name=N'dbo', @level1type=N'TABLE', @level1name=N'sysdiagrams'; GO

Note: SSMS v18 release notes