Its not a problem at all. These are called overtones and are produced with each note without actually being played. You may happen to have a better ear for them than most, but every single note you hear consists of many pitches. (The only exception is a sine wave, which is a pure tone with no harmonics. A tuning fork comes close, but still has them.) The harmonic series tells us, mathematically, what notes will sound good together and what notes will clash. It explains why a major chord sounds different than a minor chord, and many other aspects of music and harmony that we often take for granted. Each instrument, based on size, shape, material, etc emphasizes different harmonics, giving it a distinct sound, even between different models of the same instrument. (There are, of course, many other factors involved in creating a specific sound.)
The note you play is the fundamental. The most prominent harmonic is almost always one octave above the fundamental, vibrating at a ratio of 2:1, and can be quite hard to discern as every other oscillation aligns with the fundamental.
The next harmonic is often the easiest to hear as it is farther away from the fundamental, but still prominent and low enough. It vibrates at a ratio of 3:1 with the fundamental, which puts you an octave plus a fifth above. If C is our fundamental, this would be G.
We continue up the series with the ratios 4:1 (a fourth above the previous harmonic, 2 octaves above the fundamental), 5:1 (up another major third), 6:1 (up another minor third, now 2 octaves above the second harmonic), and so on (7:1, 8:1, 9:1), until they are too high for us to hear.
Notice how the most consonant intervals (octave, fifth, fourth, major third, minor third...) are closest to the fundamental. More dissonant harmonies occur further up, and are mostly imperceptible. (What actually determines how consonant two notes are is the number of pitches that overlap in their respective harmonic series. The less they overlap, the more dissonant the interval.)
Tuning based on the harmonic series is called Just Intonation and is the only way to be perfectly in tune, but the ratios create notes that are unique for each key. To avoid tuning to a new fundamental every time we change keys we created the Equal Temperament tuning system, which closely approximates pitches from the harmonic series, while keeping the distance of each half step constant (but also preventing you from ever truly being in tune!)
The human voice is incredibly good at creating overtones, since we can dynamically change the shape of virtually all parts of the "instrument" on the fly and can produce any microtone, allowing us (with lots and lots of practice) to accurately tune to the harmonic series of whatever the fundamental happens to be at the moment.
Just for fun, try singing the word "we" on a relatively low, but comfortable pitch (doesn't have to be in tune with anything). Now do it again, but REALLY slowly, shifting from the "oo" sound at the beginning to the "ee" sound at the end as gradually as you possibly can (also try to create a lot space in the back of your mouth). If done right, you should hear at least part of the harmonic series in ascending order.