Good question. I'm in NJ zone 6. My figs always grow more like a large multi branched bush than a single stem tree, so I leave all the side shoots alone. They are going to grow that way eventually. I actually anticipate this and pot up my figs from cups to 3 cuttings per 3 gallon pot. From there 15 gallon pot, and then depending on variety, they go in the ground as a good size bush. 3-5 stems and about 3-4' tall.
If you live where you can get figs to survive the winter consistently without much frost damage, than maybe you can grow yours more like a single stemmed (trunked) tree. In that case, yes trim off all the side shoots. Eventually, you will want to plant it at soil level equal to the bottom of that main top branch.
So the answer is, it depends on your weather conditions. If it ever dies to the ground it will re-grow as a bush.