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Somehow I'm not surprised that James McKee the sailor is James McCabe the policeman  Well done bbart 

Regarding James McCabe's Co. Monaghan connection- the clipping sounds as though fellow policeman accompanied the body back to Monaghan for burial (as a sort of guard of honour) and he could quite possibly be buried with family there. It's possible that one or both parents were already dead. He may or may not have had siblings.
There may well have been an announcement of his death in the local Monaghan paper but it's possible that paper/issue is not online.

Do we know what religion James McCabe was? if Protestant perhaps one reason he and Annie didn't marry.

Off to Monaghan (in a manor of speaking) to check on a few things I saw earlier 

I have just searched marriages in Belfast registration district from 1884-1889 (almost 900 of them) for anyone named Annie but didn't find anything useful. This is still this marriage I mentioned earlier- John McCabe, policeman, to Anna Meek but she's a teacher.
https://civilrecords.irishgenealogy.ie/churchrecords/images/marriage_returns/marriages_1886/10848/5960751.pdf

I had listed Annie's grave entry in Findagrave back in reply 66, and was surprised with both the grave and newspaper notice being as "McCabe".
Could it be when the McCabe name was used, it was the son John James, doing the paper work for his mother, whether it be her burial, or directory/ house lease, etc, and the entire rest of the family used McKee?

Minimal detail on Find A Grave entry but cemetery link shows address, plot number, etc. It's nice to think family are visiting Annie's grave even if there turns out to be no headstone there.
I think you are right about John James putting McCabe name down as he obviously continued to use it as his surname. He was born & baptised as McKinley in 1888, by 1893 his mother was with James McKee but in 1911 the family are McCabe so John James certainly was old enough to remember other surnames.

Found where Annie is buried! Belfast City Cemetery- Glenalina Extension (G1 334)
Annie McCabe, 12 Donnybrook St., age 69, died 19 May 1936 (matches death registration).
https://online.belfastcity.gov.uk/find-burial-records/BurialRecordDetails.aspx?RecordID=6607.10167
Unfortunately no one else listed in plot 

Though about that also but nothing, so far, explains why we can't find a marriage. Also, there's that strange mention of an unknown wife for the James who died in 1902.