A family of Microsoft word processing software products for creating web, email, and print documents.
In Word 2003 and earlier, text in text boxes was "invisible" to Word for many purposes (such as creating cross-references and TOCs), but I wasn't aware that this affected StyleRef fields. In any case, in Word 2007 this problem was eliminated for cross-references and TOCs, and I can't imagine that that wouldn't apply to StyleRef fields as well. However, I can reproduce your issue in both Word 2003 and Word 2010.
What Word 2007 and above seem to prefer to use for the purpose you describe, however, is mapped content controls (see #7 in http://gregmaxey.mvps.org/word_tip_pages/repeating_data.html); a simpler solution that might do just what you need is to use a frame instead of a text box. In Word 2003, the Format Text Box dialog has a "Convert to frame" option; unfortunately that's no longer available in Word 2007 and above. But you can draw a frame in the same way you drew the text box; the Insert Frame command is in the Legacy Forms palette of the Legacy Tools in the Controls group on the Developer tab, but you can also add the button (Insert Horizontal Frame) to the QAT is you use it frequently.
By default the frame (like the text box) will have a border; unlike the text box, which has a Shape Outline, the frame has an ordinary border, which can be removed with the No Border button on the Borders dropdown in the Paragraph group of the Home tab.