That's how I took that - different telescope. It was a telescope, rather than the telescope.
I think we may be over thinking it a bit, guys. The only on screen evidence concerning the identification of the telescope (that was given to Burnham) was holo Phillipa stating that whatever she was giving to Burnham had been in her family for centuries. There's no on screen evidence that it is a duplicate or refurbished, or anything but the telescope that belonged to Georgiou as seen in the first episode. I can see keeping dups of things you collect, but this was a family heirloom.Starfleet didn't get the telescope. They somehow got the box containing the will and telescope. Georgiou put the telescope in the box. It may not even be the telescope from Shenzou. She said it belonged to her family for a long time.
Because it is the telescope that belonged to Georgiou, (according to the holo), apparently someone grabbed it when the Shenzou was evacuated. Then, whoever went through Georgiou's stuff, likely a family member, found that it was Georgiou's wish that it be given to Burnham and sent it on.