In Islam, are theophanies possible? Can Allah visit a Muslim (or even a non-Muslim) to share a direct one-on-one message? Are testimonies of encounters with Allah common in Islam?
No. In Islam, there are no man-to-god physical/verbal encounters. While yes we do believe God revealed verbal messages through prophets, it was never really Him coming down to earth as a human or verbally speaking to people.
The only person He showed Himself to and spoke one-on-one would be Prophet Moses.
Allah says in the Qur'an 7:143
"When Moses came at the appointed time and his Lord spoke to him, he asked, “My Lord! Reveal Yourself to me so I may see You.” Allah answered, “You cannot see Me! But look at the mountain. If it remains firm in its place, only then will you see Me.” When his Lord appeared to the mountain, He leveled it to dust and Moses collapsed unconscious. When he recovered, he cried, “Glory be to You! I turn to You in repentance and I am the first of the believers.”
He won't show Himself to us humans nor reveal any more spiritual revelations to anyone else after the final prophets because they were the chosen ones. The whole divine revelations were only left to the Prophets, not people.