It would seem that Moshe already knew that God would speak to him from between the cherubim, since God explained that to him during the initial instructions on how to build the Mishkan (See Exodus 25:22). This would preclude your idea that Moshe did not know who was speaking.
Jake seemed to be on to an answer in indicating that the sentence structure is similar.
Rashi's commentary on the first verse of Vayikra may help as it refers directly to the last verse in Nasso:
to him: Heb. אֵלָיו [That is, God spoke only to Moses. This phrase comes] to exclude Aaron. Rabbi Judah [Ben Betheira] says: “Thirteen times in the Torah, God spoke (וַיְדַבֵּר) to both Moses and Aaron together, and, corresponding to them were thirteen [other] occasions [when God spoke only to Moses] precluding [Aaron], to teach you that they were not said [directly] to Aaron, but to Moses, that he should say them to Aaron. These are the thirteen cases where [Aaron was] precluded: (1) ”To speak with him…,“ (2) ”…speaking to him…,“ (3) ”…and He spoke to him“ (Num. 7:89); (4) ”I will meet with you [there at set times], etc. …“ (Exod. 25:22) All of them can be found [in the above dictum of Rabbi Judah] in Torath Kohanim (1:4). Now, [even though it was Moses who exclusively heard the prophecies,] one might think that they [i.e., the rest of Israel, nevertheless] heard the sound [of God] ”calling“ [to Moses preceding the prophecy]. Scripture therefore, says: [not ”He heard] the voice [speaking] to him (לוֹ),“ [but] ”[he heard] the voice [speaking right up] to him (אֵלָיו)“ (Num. 7:89). [This verse could have used the word לוֹ, ”to him,“ rather than such an exclusive expression as אֵלָיו, ”right up to him." However, it uses this expression in order to teach us that only] Moses heard [the Divine voice calling him], while all [the rest] of Israel did not hear [it]. — [Torath Kohanim 1:4]
אליו: למעט את אהרן. ר' יהודה בן בתירא אומר שלשה עשר דברות נאמרו בתורה למשה ולאהרן, וכנגדן נאמרו שלשה עשר מיעוטין, ללמדך שלא לאהרן נאמרו אלא למשה שיאמר לאהרן. ואלו הן שלשה עשר מיעוטין לדבר אתו, מדבר אליו, וידבר אליו, ונועדתי לך, כולן בתורת כהנים. יכול שמעו את קול הקריאה, תלמוד לומר קול לו, קול אליו (במדבר ז פט), משה שמע, וכל ישראל לא שמעו:
As the commentators indicate, it seems that there is some sort of link between these two verses. In Vayikra, God speaks directly to Moses, "right up to him", such that Moses knows Who is speaking without the actual Name of God being mentioned until several words later. In Nasso, Moses goes in to speak "to It", such that Moses is speaking with the Voice without it being mentioned until several words later. Rashi's commentary states that only Moses could hear God speak when He called in Vayikra. To an onlooker, it would thus have seemed as if Moses was talking to no one. The use of the verse in Nasso as an example would seem to imply that, in the same way, when Moses spoke to the Voice, an onlooker would have thought he was speaking to the Ark or to the Tent of Meeting.
Hope this is helpful.