Transformer inductance also changes with signal level, increasing at low frequencies up to the point of saturation, so low-frequency response could be better than a low-level measurement would indicate. And DC current also affects inductance, though this effect is reduced by gapping the core. Even push-pull transformers may have a small effective gap if the E and I laminations are inserted in blocks rather than alternating them. PA amp and organ transformers were often made this way to reduce the effect of mismatched output tubes (and it reduced manufacturing labor!).
For Jack's question, I think manufacturers tended to use the unloaded impedance ratio of the primary to full secondary (rounded off to the nearest 100 or 1000). This is known (they counted the turns) and easily measured with precision. They could have even used the impedance they were shooting for when they started the design, which ended up better with a slightly different ratio since filling all layers give the best coupling. Secondary taps are seldom in the exact 4/8/16 ratio, and even if they were, winding resistance may not be, resulting in slightly different loaded impedance for each tap. Here are the Dynaco transformers I have tested, all on 8 Ohm tap, would not be the same on 4 or 16 Ohms.