Is it permissible for a Muslim to join someone praying alone in their prayer, or is that not allowed?

There is no harm in a solitary worshiper becoming an Imam, and this is the correct view, because the Prophet, peace and blessings be upon him, transitioned from the intention of praying alone to the intention of leading the prayer when Ibn Abbas prayed with him. This is permissible in both supererogatory and obligatory prayers.