In both the Old and New Testaments, we find it common for people to have more than one name. We also see this characteristic popping up in as small a group as the 12 disciples of Christ.
We see that Thomas had the name of Didymus (John 20:24). Why don't we say "Doubting Didymus" instead of "Doubting Thomas"? It has more of a ring to it!
We find Simeon (or Simon) renamed Peter by Jesus and yet Peter is thereafter called by either name. In Acts 15:7 & 14 we see the names used interchangeably in a single chapter.
With Peter, Paul (who was Saul), and Thomas, we are told of their other name specifically referring to them.
With Matthew, we're not. We just find the account of his life given with either name dropped in (Matthew 9:9; 10:3; Mark 2:13; 3:18; Luke 5:27; Luke 6:15).
From the context, I would tend to think that the name Levi was his name from his father and Matthew was a name given him by the Lord.