I noticed when I play for example a fretted A note on the E string, the below A string starts oscillating as well. I can observe this behaviour on all other strings. Why is this? Is this intended or do I have to get my acoustic guitar repaired?
This phenomenon is called sympathetic resonance. There's nothing wrong with your instrument, in fact it's probably a sign that your instrument is in good working order!
What's happening is that the when you play the
A on the
E string, your instrument & the air around it is vibrating at that frequency (among others, but for the the purposes of this explanation the 110Hz fundamental frequency of the
A is important). This is the frequency at which the
A string vibrates when plucked. When the instrument & the air around it vibrate at that frequency the string vibrates in sympathy. You will also notice that your top
E will vibrate along with your bottom
E because of the same principal. And if you use a "just" tuning on your guitar the
B string will vibrate along with the bottom
E. In fact, the strings that align with pitches in the harmonic series of the note played will vibrate along with it! (You can demonstrate this on a piano by holding down the sustain pedal, striking a note & then damping that note with your finger & listening for the sympathetically resonating strings.)
Adding to @David Bowling's comment, quote:
Sympathetic vibrations: when you play the fretted A, the instrument vibrates with frequencies that of course induce vibrations in the other strings.
Each string has natural resonances at the fundamental, first octave, fifth above that, fourth above that, and so on. These correspond to vibrational nodes (spots with no string displacement) only at the nut and bridge for the fundamental, then with one node midway, then two nodes at 1/3 and 2/3, and so on.
Anyway, what happens is this: every vibrating frequency in the body of the instrument transmits energy into each string. However, only those harmonic frequencies a self-supporting; all other frequencies damp themselves out because the travelling energy wave, reflecting off the endpoints, is out of phase with itself and thus cancels out.