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Description
2018

Putting this into art category only because DA literature doesn't allow the picture part of it to be big enough.
I do not own the uploaded picture, only did the blue coloring and Med ctr+C ctr+V ing.

The original composition of the teams.

There are various fan approaches on explaining this, which can be classified to three large groups:
1. All mercs are indeed unique and different. Valve had no good reason to do eighteen people instead of nine, which resulted in "clones" of opposite colors.
2. The mercs are, in fact, different, but look very, very alike because of same class or other reasons. The in-game detail is too low, so we almost can't tell them apart.
3. One of the teams is the actual clone of it's opposite. All mercs are the same people.

Loop falls into the third group... but not quite.

Before the Loop, well, looped, there were only ten people, five in each team (see pic above). Blutarch and Redmond learned each other's fratricidal plans and tried to beat it by hiring more mercenaries. They already come from different countries, what literally tells us that there are very few people of this kind on the whole planet. There was little time to look up for them too, as if one of the brothers haven't hired them quick, the other would have done it. Eventually, the teams ended up having five people each.

1. BLU traditionally is attacking, while RED traditionally is defending. With an absolutely identical composition, this would not make sense. So, it can be concluded that once the team composition was not the same, which allowed them to lean toward one tactic more strongly than toward the other. Fortunately, in the menu everything is signed - we can boldly paint all offense blue.
2. Spy and Scout are a priori in different teams. This is clear from the naming: a spy is who gathers information secretly(=RED) while a scout does it "openly"(=BLU). More than that, a spy, by defenition, can get any information including tactical, while a scout is mostly about getting only tactical. This might not fully prove the unnecessity of them both being in one team, but both teams were in equal need for an informator and, most likely, both had one.
3. Same reason for two medics, except this is to explain what I did before I made AU as it is in its current state, no looking back at TF2 lore here. Fritz(BLU) and Maniac(RED) were always different people, and also they knew each other before they were hired. (It even is possible that Maniac was the one who told Fritz about an interesting job offer as he's clearly been long aware of Fritz's "misbehavior"), though, every known Loop medic clone whether BLU or RED is Maniac's.
And we're left with Sniper, who goes to BLU to balance the numbers.

A while after looping people put aside their divisions, they discover respawn, cloning, some Machine mistakes, but soon after discovering teamswitch mistake fragile peace had ended with the first Loop World War, during which teams actively tried to take over people they missed in their own teams, slowly leading them to gain equal composition. Soon there appears a concept of nine "classes", much like a strict caste system where every clone is associated with his ancestor's class, further causing class-specific weapons, clothes and other things. Though traits which can and cannot vary between clones are formed in loopians' minds spontaneously (like beards being free to have or not for anyone while sideburns are a must for Snipers only), they show strong hatred toward semi-classes for their "unpredictability", praising the same unpredictability in pure-classed. (Later, this class-limited thinking will also make loopians uncomfortable meeting people of outside the Loop and make them feel the urge to assign the outsiders to any of nine possible classes even if the similarities are close to none. Before they do so, they don't know how to treat the stranger and may behave unusual).
Uncontrolled killing/respawning/cloning cycles result in (yet) uncontrollable population growth and need to stop World War. People do not stop killing as it doesn't really much affect anything, but slow the killing spree down and go for new bases to settle... there goes a whole another story.

Description
2018

Putting this into art category only because DA literature doesn't allow the picture part of it to be big enough.
I do not own the uploaded picture, only did the blue coloring and Med ctr+C ctr+V ing.

The original composition of the teams.

There are various fan approaches on explaining this, which can be classified to three large groups:
1. All mercs are indeed unique and different. Valve had no good reason to do eighteen people instead of nine, which resulted in "clones" of opposite colors.
2. The mercs are, in fact, different, but look very, very alike because of same class or other reasons. The in-game detail is too low, so we almost can't tell them apart.
3. One of the teams is the actual clone of it's opposite. All mercs are the same people.

Loop falls into the third group... but not quite.

Before the Loop, well, looped, there were only ten people, five in each team (see pic above). Blutarch and Redmond learned each other's fratricidal plans and tried to beat it by hiring more mercenaries. They already come from different countries, what literally tells us that there are very few people of this kind on the whole planet. There was little time to look up for them too, as if one of the brothers haven't hired them quick, the other would have done it. Eventually, the teams ended up having five people each.

1. BLU traditionally is attacking, while RED traditionally is defending. With an absolutely identical composition, this would not make sense. So, it can be concluded that once the team composition was not the same, which allowed them to lean toward one tactic more strongly than toward the other. Fortunately, in the menu everything is signed - we can boldly paint all offense blue.
2. Spy and Scout are a priori in different teams. This is clear from the naming: a spy is who gathers information secretly(=RED) while a scout does it "openly"(=BLU). More than that, a spy, by defenition, can get any information including tactical, while a scout is mostly about getting only tactical. This might not fully prove the unnecessity of them both being in one team, but both teams were in equal need for an informator and, most likely, both had one.
3. Same reason for two medics, except this is to explain what I did before I made AU as it is in its current state, no looking back at TF2 lore here. Fritz(BLU) and Maniac(RED) were always different people, and also they knew each other before they were hired. (It even is possible that Maniac was the one who told Fritz about an interesting job offer as he's clearly been long aware of Fritz's "misbehavior"), though, every known Loop medic clone whether BLU or RED is Maniac's.
And we're left with Sniper, who goes to BLU to balance the numbers.

A while after looping people put aside their divisions, they discover respawn, cloning, some Machine mistakes, but soon after discovering teamswitch mistake fragile peace had ended with the first Loop World War, during which teams actively tried to take over people they missed in their own teams, slowly leading them to gain equal composition. Soon there appears a concept of nine "classes", much like a strict caste system where every clone is associated with his ancestor's class, further causing class-specific weapons, clothes and other things. Though traits which can and cannot vary between clones are formed in loopians' minds spontaneously (like beards being free to have or not for anyone while sideburns are a must for Snipers only), they show strong hatred toward semi-classes for their "unpredictability", praising the same unpredictability in pure-classed. (Later, this class-limited thinking will also make loopians uncomfortable meeting people of outside the Loop and make them feel the urge to assign the outsiders to any of nine possible classes even if the similarities are close to none. Before they do so, they don't know how to treat the stranger and may behave unusual).
Uncontrolled killing/respawning/cloning cycles result in (yet) uncontrollable population growth and need to stop World War. People do not stop killing as it doesn't really much affect anything, but slow the killing spree down and go for new bases to settle... there goes a whole another story.