There are different ways
First and foremost, aircraft radars have receivers, which will also receive the transmitted pulse from the other radar. In fact, that is how radar jamming works - you deluge the other radar with power on the same band / wavelength, and it will fill the display with clutter.
With that said, radars have a bunch of characteristics that makes it somewhat easy to identify. Frequency, PRF (pulse-repetition frequency), waveform, etc.
The bands at which radars operates are regulated, and various agencies - military and otherwise - will pick up radar transmissions. These tiny DIY projects aren't a huge "threat" in that sense, but if you're active anywhere near a radar installation, and especially military ones, it could land you in hot water, real fast.
I'm not sure about Finland, but where I live, civilians for example are not allowed to fingerprint or build fingerprint databases of military radars. In fact, the military is the only ones allowed to do so.
The power-scaling makes these things go from DIY to non-DIY real quick.
Would be cool to see this guy build a phased-array (radar)
Source: Worked with radars