What you are experiencing is the physics of sound. The same program twice can be the cause of cancellation of frequencies... What you are saying sounds "strange" is most likely due to some cancellation of frequencies.

This is why this is usually not done; without arranging the pan position or delaying one of the channels a bit more than would occur in a synth engine trying to play the same program twice. Doubling a sound is better done using delay effects than how you are doing it.

Your way:
Possible strange phasing and cancellation possible
Uses twice the amount of polyphony

Using Effect processing:
Avoids strange phasing sound, and avoids cancellation of some frequencies
Avoids using double the polyphony

Doubling the sound does not double the level output, sound stacks up logarithmically (it would take ten times stacking to double the sound output).

In order to thicken or make a richer sound, recording engineers often us a very short time delay, without any LFO movement, just a stright short delay, very, very short and send the delayed signal to the opposite side of a stereo mix. The sound reaching the right ear versus the sound reaching the left ear gives the brain a sense of size and distance. So using delay and pan position you can make your string sound bigger, richer, thicker... And avoid cancellation and polyphony issues.

What you are experiencing is the physics of sound. The same program twice can be the cause of cancellation of frequencies... What you are saying sounds "strange" is most likely due to some cancellation of frequencies.

Interesting.
Am I right in guessing that this means I can't even layer two similar voices (i.e. Almighty and Small section)? Or does this only happen with the exact same voice?

Back to my original purpose: I wasn't trying to double the sound, but rather I have a portion of a song where some strings hold a chord, while others play a melodic passage on almost the same notes.
I thought about using the first voice for the chord, where I would hold it with the sustain pedal (of course I transpose that voice so that I actually press different keys), while the second voice is for the melodic passage.

What's the proper way to this?

Interesting.
Am I right in guessing that this means I can't even layer two similar voices (i.e. Almighty and Small section)? Or does this only happen with the exact same voice?

It happens in the unnatural situation where the sound is exactly identical. This is the exception in nature - that two musical instruments would sound exactly the same, in phase, pitch, and every aspect. The ear/brain can detect the slightest variation. And when it doesn't we describe it as "strange" as in unnatural, just as you did.

Almighty and Small Section, are not identical (not even close), your ear/brain will immediately hear and recognize this. In the real world, several musicians attempting to play together actually do not play exactly together. This is the nature of music. If you have two string players reading the same chart, there may be brief moments when they do actually reach exact duplication of pitch and phase, those moments are brief, and fleeting... (And tickle our senses...) mostly they are identifiable as an ensemble because of the subtle differences. Learn to listen closely to musical signals.

I say that based on my own experience... As a musician, I of course, listen to music with a typical musician's sense, but studying to become a recording engineer while in my twenties, taught me to hear music with a different focus. Particularly when it came to positioning the sound in the stereo panorama, phasing, pitch, and the illusion of placement.

Back to my original purpose: I wasn't trying to double the sound, but rather I have a portion of a song where some strings hold a chord, while others play a melodic passage on almost the same notes.
I thought about using the first voice for the chord, where I would hold it with the sustain pedal (of course I transpose that voice so that I actually press different keys), while the second voice is for the melodic passage.

There is no proper way to do this... But you have to try some different things until you find one that works for you.

I would use an entirely different sound (based on what you've described) reason: strings holding a chord, are different from strings articulating a musical melodic phrase. Of course, I'm at a disadvantage because I can't hear the music in question, you will need to decide, but having access to different articulations is a key "new" feature added to hardware Yamaha synthesizers with the XS series. It purpose is to allow you to do exactly what you are attempting, and to do it in realtime as part of your interaction with the synthesizer engine. It referred to as expanded articulation control.

For example, find the Voice: Medium Hall
Played normal you get arco (bowed) strings, which you can hold with a sustain pedal.
Press [AF1] to select a quick articulated bow stroke
Press [AF2] to select pizzicato (plucked) string articulation, you can play across the entire keyboard using either of these articulations, even triggering the notes of the sustain pedal held keys!

No these may not be the articulations you require, but with XA CONTROL (eXpanded Articulation Control) you can go from long bowed to quick bow strokes for executing a string phrase as you may require. Sorry, without hearing what you are doing I can only point out broad examples.

But XA Control, is what Yamaha introduced to do exactly what you are attempting, you will want to substitute the String sound (Element) that provides you with the articulation you require to execute your melodic passage.

Thank you for your thorough explanations, I'm astonished!

As I've just wrote in another thread, these options require time I don't have now 🙁
But I'll be back as I find time for it 😉

Thank you again.