"Incense" (or frankincense) was a costly substance that created a fragrant smoke when burned. As the verse indicates, in this case it symbolizes our prayers rising to God.
The phrase, "a Deity nigh" refers to the Christ Child himself, who was "near" (nigh) from the perspective of the three Kings who had traveled to worship him and offer Him gifts. Nigh is an old word for near. And Deity means divine, or God.
"Owns" in this case probably does not mean the obvious fact that Jesus now "owned" some incense since that was one of the gifts. "Own" can mean to have authority or mastery over something. The smoke from smoldering incense symbolized the prayers of God's people rising to heaven. And Jesus, as God's Son, rightfully has power to receive our prayers and answer as He sees fit in His love and wisdom. He is the One who can hold our prayers and own the outcome.
So incense (a symbol for prayers) owns a deity neigh.
One can picture the scene with Joseph, Mary, the baby Jesus, and the wise men, or Magi as they were called.
To paraphrase the line, "Incense--representing our prayers--belongs to this child before us, who is God in the flesh.
Frankincense to offer have I;
Incense owns a deity nigh;
Prayer and praising, voices raising,
Worshiping God on high.