How is David playing a melody from an out-of-universe soundtrack?

How is it possible that the melody he composes is identical to the melody in Prometheus (2012)'s theme soundtrack (song name Life)?

Actually, we should turn that question around: why do you think it's impossible?

If there is no proof of something being impossible, then it must logically be possible. Please do note that "possible" does not mean the same thing as "likely" or "expected".
Many stories revolve around the fact that something possible-but-improbable happens. Some random examples:

• Someone shoots a bullet, it deflects and bounces around and ends up killing the shooter. That is obviously a one-in-a-million shot, but it would be wrong to say that it is impossible to ever happen.
• A hero jumps out of a moving vehicle (at high speed). It's very likely that he gets hurt. However, it's not impossible that he manages to walk away from it. It's just very unlikely.

It would be wrong to call these plot holes or movie mistakes. It would only be a movie mistake if you can prove that it couldn't have happened (= proving that it is impossible).
E.g. the bullet reflected off of a piece of paper (impossible!), or the hero jumped out of a car and smacked face first into a concrete wall (impossible to walk that off!)

A bit more specific to your case, I can think of another example.

In Battlestar Galactica (the 2003 remake), Anders suddenly remembers a song that he himself had written. This song is "All Along The Watchtower", released by Bob Dylan and covered many times, most notably by Jimi Hendrix. However, there is no relation between the in-universe song and the out-of-universe song.

The song's out-of-universe existence does not impact the plot in any way.

However, BSG's premise heavily relies on the implication that "all of this has happened, and will happen again" (it might be slightly different the next time, but it boils down to the same principle).
Because All Along The Watchtower has been covered so many times; you can see the same premise occur in that song: it has been repeated many times, with slightly different version, but it ends up being the same song with the same lyrics.

That is a thematical connection that has no bearing on the in-universe plot, it is merely an easter egg for the viewers.

Conclusion

The same seems to be true for Alien Covenant. This song is original in-universe, even though it isn't original out-of-universe. The fact that it isn't original out-of-universe has zero impact on the plot, and is therefore irrelevant in-universe.

Unless a connection is made between the two, there is no reason to assume that this is done for any reason other than an easter egg or quick nod to the viewers.

If I extend the principle behind your question, I can show you some clear examples why your question's implied logic is inherently flawed:

• How is it possible that people in Middle Earth have swords, when Middle Earth is not our world?
• How is it possible that Let It Go is popular on Spotify, when it is originally sung by a fictional character in a movie?

The answer to these examples is the same as your question:
Just because one of the universes (fictional/real) has done something, doesn't make it impossible for the other universe (real/fictional) to do the same thing for a completely different and unrelated reason.

You have to think of the context where the song was last heard, it played while David observed the Engineers Orrey. And stopped once the projection stopped. Even when David appears at the engineers planet there is a trumpeted fanfare which is part of "Life."

We cannot say the song did not exist in the universe of Prometheus and Alien: Covenant, only David and the engineers presumably have heard the tune.

About that scene: The Orrey was based on "The Philosopher Lecturing On The Orrey" by Joseph Wright of Derby. A painting highlighting and romanticising the age of enlightenment, when science and fact based beliefs started to rise to prominence over Christian values and beliefs.

Even the art style chiroscuro was previously reserved for the depictions of Christian stories or Greek Gods.

In Advent David discovered that Shaw's DNA/Human DNA unlocked special properties with the mutagen and helped in developing the perfect organism. To me David's elegy for Elizabeth may be a sign of the discovery made when experimenting on her, just like the discovery of the Orrey. New worlds and opportunities to learn. The tune is not original, but a copy. Like most of the Easter eggs in Alien: Covenant.

May be far fetched thought process, but I have analysed both movies quite in depth. This is just my opinion.

Actually, it is out of universe.

For those that follow Star Trek: ST-NG-S06E-21: Picard follows his love into a new area, where both play this tune, not quite this tune, but the tune is based upon that "folk tune". Check for yourself.

I believe the creator made a wink to this episode for some reason, leaving an western egg, if you like.

I already noticed this in the first movie, but I was uncertain until I heard the flute version played by David.