Careful with terminology -- the port impedance may be very different from the expected load impedance. A typical case being open-collector outputs, which are quite high impedance (kohms) with a little capacitance, and that's about it. The expected load might be whatever, but the impedance is high. Or the opposite direction (low Z) for voltage followers, of course.
The maximum power transfer theorem, for example, still applies as ever, but only while linear -- you will indeed get maximum gain into a high (kohms) load in such a case (collector output), but the power handling will be minuscule. In this kind of case, the match is instead chosen for maximum power, or bandwidth, and the impedance is dealt with however it must be dealt with. Often, this is by placing a load/termination resistor there, so the source is reasonably Norton-equivalent and you can use your usual filter prototypes with it. Or if more power gain is desired, then a one-port-open filter prototype must be used.
Put another way: the voltage gain (or corresponding power gain) may be correct for the resistor values shown, but need not be correct for other resistor values, even though the same resistor ratio is used.
If the port impedance is not specified (or something equivalent, like s22), you probably have to measure it to be sure. Or, of course, stick to the prescribed methods and values, and, perhaps, buffer/amplify it at your earliest convenience to get the impedance into something reasonable and expected?
Or, put still another way: I don't know. (Helpful, right?...) But, I can give this background, which can allow you to figure it out with a little analysis. For example, if the change from balun to resistor divider gives 30dBV insertion loss*, then you should be able to calculate backwards and figure out the output port impedance, because one way it's loaded with 300 ohms and has X gain, and the other way it's loaded with ~1.7kohms and a divider and has X - 30dB gain.
*Or, whatever it's measured relative to. The screenshot doesn't say. Or, actually, if it's voltage gain, is it not just the literal voltage between the port pins (as loaded), versus the output connector? If that's what they're measuring against, then the resistor divider has a ratio of 1/2 * (53 / (820 + 53)) = 0.0303, the 20*log10 of which is damn close to -30.36dB. (Note the factor of 1/2, because you measure the voltage of a differential port, well, differentially; about half is discarded through the spare resistor, so, 1/2.)
Tim