In what ways does Borges play with the Englishman's identity in The Shape of the Sword?In other words, at which points in the story is the man's...

Ambiguous refers most often to that which has an unclear double meaning. It is derived from the Latin word ambiguus meaning "having double meaning, shifting, changeable, doubtful." Another, less common, usage of ambiguous means doubtful; uncertain; difficult to comprehend.

By these definitions, this question is a bit ambiguous.... If you mean where in the story The Shape of the Sword (Borges)are we torn between choosing one of two hinted identities, that only occurs at the end of the story where the Englishman says,

"From one of the general’s collections of arms I tore a cutlass, with that half moon I carved into his face forever a half moon of blood. Borges, to you, a stranger I have made this confession;"

and then again where Borges and the Englishman say,

"Here the narrator stopped. I noticed that his hands were shaking.
“And Moon?” I asked him.
“He collected his Judas money and fled to Brazil...;" and finally when the Englishman stammers, "Don’t  you see that I carry written on my face the mark of my infamy?"

If, on the other hand, you mean where in the story is the Englishman's identity uncertain and doubtful--which fits the idea of Borges playing with the Englishman's identity--that occurs in several places throughout the text. The first instance where Borges teases the reader with allusions to secrets relevant to identity is where Borges, as the narrator, says,

"I understand that the Englishman resorted to an unexpected argument: he confided to Cardoso the secret of the scar."

The sentence after this, "The Englishman came from the border, from Rio Grande del Sur; there are many who say that in Brazil he had been a smuggler," lights up in retrospect as a clue at the end of the story when the Englishman says Moon reportedly went to Brazil; it's foreshadowing. 

The second place Borges teases us with hints about identity is when the Englishman is waxing philosophical while saying,

"This frightened man mortified me, as if I were the coward, not Vincent Moon. Whatever one man does, it is as if all men did it...I am all other men, any man is all men."

Then Borges's final tease comes in a line also quoted earlier, but then in reference to the meaning of ambiguous reflecting unclear choices. It is:

"From one of the general’s collections of arms I tore a cutlass, with that half moon I carved into his face forever a half moon of blood."