Who does the Monitor think the Master Chief is?

Guilty Spark refers to The Chief as "Reclaimer". This is an ambiguous title, not specific to any particular person.

A Reclaimer is an individual who, whether by chance or by precedent, has been placed in charge of activating a Halo super-weapon. [...] The exact qualifications of a Reclaimer are not entirely clear, but only a human or Forerunner can be a Reclaimer. Reclaimer has often been used by Forerunner AI and Huragok as a general term for humanity, although not all humans appear to qualify.

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In the original game, the context is ambiguous, making this a little vague as to what or who he believes he is talking to. However, in the extended canon, it is made a bit clearer:

The arrival of the Covenant on the ring instigated Installation 04's genocide and the final Flood outbreak. Immediately putting long dormant contingency measures into action, Guilty Spark began to search for a Reclaimer and discovered UNSC Staff Sergeant Marvin Mobuto. Using Halo's teleportation system, Guilty Spark brought Mobuto to the Library in an effort to retrieve the Index. Ultimately, although he fought hard, and endured the Flood almost to the Index, the ill-equipped Marine failed and was killed, forcing the Monitor to seek out another Reclaimer. John-117 discovered the body of Mobuto (which was mangled beyond usage by the Flood) during his own journey through The Library, though it can't be seen in-game, and marveled at the Staff Sergeant's ability to get as far as he had with less optimal equipment.

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So what we can take from this is that 343: Guilty Spark does not see the individuals, or even distinguish between two individuals, particularly in the role of the "Reclaimer".

"Ah" 343 Guilty Spark said, peering down over the Spartan's shoulder. "The other Reclaimer. His combat skin proved less suitable than yours."
The soldier looked up over his shoulder. "What do you mean?"
"Is this a test, Reclaimer?" The Monitor seemed genuinely puzzled.

^{Halo: The Flood by William Dietz p. 244}