Staff label text is not the same as a text object or text in a text frame - you can’t select it. Staff labels take their alignment from the paragraph style only, never from the alignment options in the Edit Instrument Names dialog, but you can use the other options to customize the instrument names that appear in staff labels, such as making individual words italic, such as “for rehearsal only”.
The text editor options that appear when you add Shift-X/Shift-Alt-X text or edit text in text frames give you full control over the selected bit of text. Paragraph styles control the formatting of all text that uses the corresponding paragraph style. Header text (and by that I assume you mean the text in the text frame at the top of e.g. the Default master page) has all the same text editor options available as any other text frame or text object.