If the inputs are short circuited. Impedance is low. Why is the noise simlar to high impedance inputs?
it depends on amplifier type, some of them have higher noise figure with low source impedance, some of them have higher noise figure with high source impedance.
Also note that short circuit is a kind of magnetic loop antenna, which receive a lot of noise from nearby electronic equipment and mains line. For a clean experiment, you're needs to put your short circuit and input frontent inside a good shielded enclosure and put it away from any wires or electronic equipment for at least several meters. It should be powered from a battery to make sure that there is no common mode currents through wires or RFI received through near field coupling.
If you want to minimize noise, it's better to use shielded terminator with impedance that is well matched with your device input impedance. If your device has high impedance input (such as 1 MΩ) you can leave it open. But for low impedance input (such as 50 Ω or 600 Ω) it's important to use terminator instead of open input.
As I said before, measurement for so low signals (about 1 uVrms) is not so easy and requires special measures and measurement methods. Using proper AWG here is not enough, because every wire geometry, shielding and electromagnetic environment have significant effect on measurement results.
Common issue with obtaining microvolt signal level is leakage from signal source (through ground, through signal wires, or with near field coupling) or even through attenuator. This is why it worth to buy proper shielded cables, good quality attenuators and good AWG with low unwanted emissions to avoid such issues.
For example, when my PSG9080 AWG output is open and disabled (nothing is connected to it), I still can receive it on nearby VHF receiver which has MDS sensitivity about several microvolts...
As you can see, when you're working with microvolt signal level, the signal path is not obvious and it can bypass your attenuator through near field coupling. Especially if you're don't use shielding at all, like in your resistive divider on breadboard. This is why you can see the same amplitude for expected 1 uV and for 10 uV. It just means that you receive not signal from your attenuator, but signal leakage which bypass your attenuator.
Another point here is to use proper attenuator. Not a simple resistive divider which output impedance depends on divide ratio. Your attenuator needs to have proper input and output impedance. You're needs to use Pi or Tee attenuator, which can be calculated here for example:
https://www.everythingrf.com/rf-calculators/pi-attenuator-calculatorFor example if you want voltage ratio 1:100 attenuator, this is 40 dB attenuation. Pi attenuator for 50 Ω impedance needs R1=51 Ω and R2=2500 Ω.

But if your device input has different than 50 Ω impedance, you're needs to recalculate it. And take into account your cable characteristic impedance.
To reduce possible leakage it's better to split 40 dB into two 20 dB attenuators connected in series.
Also you're needs to take into account required bandwidth. By adding low pass filter you can cut off significant power of high frequency noise, so your measurement noise floor will be better.