During the pandemic last few years, the Infrared thermometer is widely used to measure the body temperature.
However, the fundamental question or the question in the first place is, why do we expect the radiation from our human body to be black body radiation?
As far as I know, the radiation of the earth is not a black-body radiation. See the link here
https://seos-project.eu/earthspectra/earthspectra-c02-p17.html
IR thermometers are empirically calibrated by taking a reading off the intended surface while that surface is instrumented with a precision thermocouple. The output of the IR device is then adjusted to match the output of the thermocouple. IN this way, the nonideal emissivity of the surface is accommodated.
IR thermometers for industrial use contain a manual adjustment feature so that this calibration can be performed for any desired surface.
The earth is approximately radiating out infrared radiation due to it's temperature? Anything that radiates energy due to it's own temperature is modelled to be blackbody. I'm not sure if it is actually a perfect blackbody object (most likely not) but the Stefan Boltzmann law is typically used for any object radiating out heat. This can be partially explained with some quantum mechanics and stat mech of photons in a cavity. A good example of how we are blackbodies (or at least modelled as such) would be infrared cameras observing our "heat signatures".
Edit: I've just relooked at the link. There are other ways the Earth is radiating EM waves as well. Albedo (the whiteness of the Earth caused by the ice mainly) helps to reflect light from the surface for example. Factors like this could be causing the fluctuations in the graph in the link.
Regarding the title: "When do we expect blackbody radiation", the answer is everywhere (emissivity notwithstanding).
At our maximum temperatures (tera-kelvin in heavy ion collisions), we get thermal pions radiated (I'm not sure about electromagnetic radiation--the timescales may be too short).
Ultra dense systems such as core-collapse supernovae (100 GK) radiate thermal neutrinos.
From there we have hard and soft X-rays in the various stages of nuclear fireball formation (100 MK - 285 kK)
Then lightning are arc-welding in the tens of kilo-kelvin emit UV.
The Sun is our best known visible-spectrum emitter (5772K). Vis. cuts of at the Draper Point (798K).
Then IR ofc, and from 300K down to 150K is the domain of microwave radiometers for weather and atmospheric profiling.
Down to 2.72548K, the CMB.
In microwave radiometery, emissivity plays a huge role in recovering physical properties of the surface and atmosphere, and in those systems, we speak of "brightness temperature", $T_B$ at a specific frequency ($\nu$), which satisfies:
$$ B_{\nu}(T_B) = \epsilon(\nu)B_{\nu}(T_{\rm physical}) $$
where $B_{\nu}(T)$ is the blackbody spectral radiance.
Spaceborne systems (ATMS, SSMIS, GPM, AMS, ...) report data as $T_B$, and users then apply their models to recover $T_{\rm physical}$ and $\epsilon(\nu)$
Regarding medical thermometers, idk how they are calibrated, but The Web says the emissivity of human skin is $0.98$, which is an $11^{\circ}\,$F correction at $98.6^{\circ}\,$F so it definitely matters.
Edit: since the tag says "atmospheric science", the standard is
https://www.researchgate.net/publication/299103021_Microwave_Radar_and_Radiometric_Remote_Sensing
Regarding the IR plot in your link, these system are not looking at total blackbody radiation; rather they look at specific frequencies tied to molecular absorption lines (water vapor and O$_2$ in microwave), or at atmospheric windows to get a clear view of the surface.
Near the absorption line(s) you're only seeing one optical depth into the atmosphere from LEO, so the higher the absorption, the higher the emission source, and of course: the higher up you are, the colder the atmosphere. So even if the emissivity is near 1, and the radiation is blackbody, the overall spectrum is going to be modulated because different frequencies are looking at different layers along the same line-of-sight.
Microwave systems will put several channels at the same frequency near an absorption line, and just vary the bandwidth vs the channel, thus, each measurement at the same $\nu$ (different $\Delta\nu$) will see different depths into the atmosphere, so that a temperature profile can be extracted from a single scan.
