It potentially affects every concentration spell...
...and how that interacts with every concentration spell, and how that interacts with every concentration spell, etc.
There are currently 237 concentration spells on D&D Beyond. Of those spells, 185 are 1st-5th level spells or cantrips, meaning a single character could have any two of them (via multiclassing). If you only want to check how any two of those 185 interact, you'd need to check at least 17020 two-spell combinations.1 And that ignores high and low level combinations, like a Wizard 17 / Cleric 1 using shapechange and shield of faith.
And don't forget - that number increases every time WotC releases another book.
"But what if I restrict things?"
The important part is how you restrict things. The goal is to reduce the number of combinations you'll need to check. Here's some examples:
Choose one 1st level spell from your class' spell list. That spell no longer requires concentration for you.
There are 29 spells that qualify for that. Each would need to be tested against every other concentration spell in the game. That leaves 6467 combinations to test.2
Bless no longer requires concentration for you.
Bless would need to be tested against every concentration spell (including itself), leaving 237 combinations. That's still a lot, but way less than the less restrictive version.
As a Cheese Domain cleric, you can concentrate on two spells as long as both spells are from your domain list.
If all 10 of the Cheese Domain spells normally require concentration, then they'd need to be tested against each other. That brings it down to 55 combinations.1
If you are concentrating on bane, bless, or shield of faith, casting a different spell from that list does not cause you to lose concentration on the first. You may not have more than two of these spells active in this way. If you lose concentration, both spells end.
This only requires checking 3 combinations1 - a far more reasonable number. Furthermore, that number doesn't increase as new spells come out.
Long-duration spells make things even more complicated
In the previous sections, we were just considering two-spell interactions. If you start a fight without any spells running, then it'll take two rounds (in most cases) for you to get two concentration spells going. However, spells with a duration longer than 1 minute are often cast before the fight even starts.
Allowing a 9th level cleric to cast shield of faith (10m duration) or spirit guardians (10m duration) without requiring concentration, for example, means that both spells plus holy weapon (1h duration) could easily be active before the cleric even opens the door! That's triple the amount of active spells for no action at all!
The designers specifically advise against it
The DMG says:
Beware of adding anything to your game that allows a character to concentrate on more than one effect at a time [...]. Rules and game elements that override the rules for concentration [...] can seriously unbalance or overcomplicate your game.
This isn't just a cop-out answer. There are very few features of the system that are called out in this way. That likely makes concentration a fundamental part of the design of the system.
The meta reasons
In my experience, newcomers to 5e are often frustrated by the concentration rules. They take damage and lose a spell, or they can't shield of faith the fighter without losing spirit guardians. They want to avoid that frustration by changing or removing concentration.
What I'm saying is - concentration changes are frequently (but not exclusively) suggested by less experienced players. When people say "don't change concentration", they usually mean "changing concentration requires extensive knowledge of the system; if you're asking non-specific questions about it, you probably don't have the knowledge required, and there's no way we can convey that in a post".
I'm not saying that reaction is right, but people are good at picking up on patterns, and this is one of them.
1 Calculated as a combination with repetition, which I think is the right math here. This accounts for bless + shield of faith being equivalent to shield of faith + bless.
2 Calculated with a simple product (237 × 29), minus the duplicates. Each spell after the first has to check one less combination (since it would be a dupe), meaning the total number of dupes is the sum of numbers between 1 and 28: 402. That leaves 6873 − 402 = 6467.