According to most mefarshim (commentators), it means that Moshe Rabbeinu equated himself with Yehoshua.
The Tur HaAroch writes:
משרת משה מבחוריו, “the servant of Moses from his youth.” Ibn Ezra writes that some commentators understand the word מבחוריו to mean that Joshua was one of Moses’ most devoted servants, whereas others view the word מבחוריו as if it were glued to the words משרת משה, meaning that he had been a servant of Moses from his earliest youth. This latter explanation cannot be correct as what is described here occurred only in the second year of the people’s being in the desert, and we have no evidence that Joshua who was only about 24 years younger than Moses already served in that function in Egypt. I believe that the proper explanation is that Joshua was one of several men who attended to Moses’ needs.
In Avot D'Rabbi Natan 27:4, it writes:
Everyone should learn from Moses our teacher, who said to Joshua (Exodus 17:9), “Choose men for us.” He did not say: Choose for me, but “choose for us.” This teaches that Moses treated Joshua like an equal, even though Moses was the master and Joshua was his student.
From this we can learn that this pasuk means that Moshe Rabbeinu equated Yehoshua with himself. This is echoed in the Midrash (Mekhilta D'Rabbi Yishmael 17:9).
Not completely relevant, but see what the Shelah HaKadosh (Torah Shebikhtav, Beshalach, Derekh Chayim 13) writes:
בחר לנו אנשים. Rashi sees in the word לנו proof that Moses equated Joshua with himself in this task. The Rabbis deduced from this a rule that a teacher should always be as concerned with the honour of his student as with his own. The honour of someone of equal status to oneself, however, should be placed on a par with the honour of one's teacher. The source for this is found in Numbers 12,11 where Aaron, Moses' older brother, addresses him with the words: בי אדוני, "Please my lord!" The honour of one's teacher should be treated as equivalent to the honour of G–d Himself. We derive this from Numbers 11,28 where Joshua pleads with Moses to kill Eldod and Meydod for prophesying the death of Moses. The words used are: אדוני משה כלאם. Joshua felt that insubordination against his teacher Moses deserved the same penalty as insubordination against G–d Himself. We can now understand why we are taught in Avot 4,1: "Who is truly honoured? He who honours G–d's creatures."