All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad is His slave and Messenger.
It is permissible to be named with more than one name. The evidence is that the Prophet had many names as confirmed in authentic narrations. Therefore, this is enough evidence to convince your friend. For more benefit, please refer to Fatwa 87307.
We do not know where he got the evidence that having more than one name necessitates being reckoned more than once. Judging a person more than once due to him having more than one name is probably not done by human beings in this world, let alone this to be done on the Day of Resurrection by Allaah, The All-Knower and The All-Wise.
Allaah informed us that He does not wrong anyone on the Day of Resurrection. Allaah Says (what means): {And We place the scales of justice for the Day of Resurrection, so no soul will be treated unjustly at all. And if there is [even] the weight — of a mustard seed (824), We will bring it forth. And sufficient are We as accountant.}[Quran 21:47]
For more benefit, please refer to Fatwa 88711.
Allaah Knows best.