Should I finish teriyaki sauce with butter?
I would not recommend doing so.

The technique you describe is known as monter au beurre, which as you might guess from the name is a common finishing method for sauces in the French culinary tradition. It translates literally to "mount with butter" and is used to provide a rich, velvety mouthfeel to both white and brown sauces.

By contrast, teriyaki sauce originates in Japanese cuisine, which is a whole different tradition. Like Chinese cuisine and other east Asian cooking, Japanese cuisine doesn't make use of much dairy, possibly due to a higher prevalence of lactose intolerance. Teriyaki sauce is generally a glaze that gets its richness from a high proportion of sugar and can be thickened with cornstarch (as seen in this fairly representative recipe).

You could try mounting a teriyaki sauce with butter, but my suspicion is that you might cause some of the sugar in the glaze to seize and crystallize by introducing cold butter (as typically recommended when mounting) and it would not be common in the culinary tradition that teriyaki comes to us from. So you can skip this step without causing any harm to your sauce or offense to any French or Japanese bystanders.