According to varius meforshim (such as Rabbi Sorotzkin, Rav Hirsch, etc) all shepherds had their own staffs. Also, a staff was a symbol of authority and every major person had his own. See the statement by Yaakov or Yehudah and Tamar as an example.
The magicians also had their own staffs as well which they used when performing magic (as when Moshe and Aharon first came to Par'o).
This was also stated in that way to show that there was nothing "magical" about the staffs, as opposed to what the magicians tried to convince everyone. Even nowadays we have stage magicians using a "magic wand" to perform their tricks.
Rav Hirsch Va'eira 7:19
Your staff, not the staff of Moses, the implement does not matter,
there is no magic in the staff, it was Hashem's action.
UPDATE @Danny Schoemann suggested that I incorporate the answer from Did Aaron's staff turned into a snake or a crocodile? into this answewr. There we see that Aharon specifically threw down his staff to turn into a "tanin" at the initial confrontation with Par'o. There again we see that the particular staff itself did not matter.