Any physical sound that has a sense of pitch(a "frequency") is composed of a root/base/fundamental frequency that we feel as the foundational pitch.
Now, it is proven that these physical sounds all have a special mathematical relationship between the root and the other "frequencies" that are heard.
The relationship is very easy: it is always an integral multiple of the fundamental. This special series is called the overtone series.
e.g., suppose you "hear" a sound that you estimate to be 500hz(say you record and then slow down and count the oscillations). Then, that tone will have, generally, other frequencies with it but, if it be a musical pitch, they will be the frequencies 1000, 1500, 2000, 2500, 3000, etc...
"Real world" pitches generally are not perfectly mathematical and other non-musical sounds could be mixed in. Regardless, the mathematical theory is highly accurate for most musical sounds. The strength of these overtones give the character to the sound we hear. A volin playing C and a guitar playing C both have a fundamental pitch of C and all the overtones are in both sounds... only the relative strengths of those pitches are different.
Now a chord is multiple pitches together. Each pitch has its own "overtone series".
For a tuner to recognize chords, it must resolve each fundamental of the chord. To do this, all frequencies must be found(through Fourier Analysis, say), then those frequencies must be tied to some overtone series.
This is not difficult to do mathematically but difficult in practice. The more notes or the more layers of instruments, the harder it is because the complexity goes exponentially.
So, yes, it could theoretically be done but no one has yet created a real device to do so. Some are working them. Some tuners now exist for guitars(polytuner, etc). At this point they generally are easily confused but anomalous inputs and definitely do not work in general.
The reason why your simple tuner does not recognize differences between major and minor is probably that it looks for the lowest note(that is easy to do) and just assumes that is the root. Since it excepts you not to play more than one note, and you do, you can't expect it to make sense. e.g., play a first inversion A chord and it should give you C, etc.
Since general music is much more complex you won't find any general analysis device or app(not yet, anyways). E.g., Say you play A5, is it a major or minor chord? it depends on the music that came before and what comes after. if you play C A5 Dm then it is an implied Am and will sound minor. If you play F#m A5 D then it is an implied A and will sound major. The tuners would have to listen to the music to figure this out. Currently only the human mind does this well and it involves training ones ear to know what it is hearing. Eventually someone will develop a device that is capable but I haven't seen anything that even comes close yet.