Unable to save database diagrams

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;
GO
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