Is there any ancient god that was named Alexis?

No, there was no Greek or Roman god or demigod named Alexis. The name Alexis is of Greek origin, and was originally a male name, meaning "defender".

There is a Saint Alexis-Falconieri in Catholic mythology. He and six of his friends left the tumult of Florence for Mount Senario to live a quite life as the "Servers of Mary".