If I was to tweak concentration rules I would probably do it through feats which allow you to concentrate on more than 1 spell at a time, with a downside which makes it dramatically easier to fail if something tries to interrupt your concentration.
I would do this, too, but not as a feat. I'd just let anyone concentrate on two things, at a +5 to the DC of Concentration saves, or three things, at a +10 to the DC. And if you fail a Concentration save, you lose ALL the spells you're concentrating on.
The reason I would do it as a feat is because concentration exists for a reason (mainly balance), but unlike most decisions which are made for balance, I actually kind of like it because mechanically, the idea of a concentration spell makes sense. The main issue with concentration is it creates a situation where it reduces the variety of concentration spells down to 1. There are lots of concentration spells, but most of the time, there will be 1 spell which is better than all the others and so you will end up casting that concentration spell over and over again. This becomes more pronounced the higher level you become, as you have access to more and more spells and the higher level spells are a lot better than the lower level spells, so pretty much all of the lower level concentration spells become dead weight which would be better used as a damage spell.
The problem with allowing casting multiple concentration spells is fairly obvious - It breaks action economy. Now, I am not a balance purist, so it doesn't go too much against my personal philosophy to do so, but I do think if we are going to deliberately unbalance a system we should minimize the damage. Making it a feat (maybe a feat which can be taken multiple times) goes some way towards minimizing the damage by forcing players who want to concentrate on multiple spells to sacrifice a feat in order to do so. It does create a secondary problem that maybe this 1 particular feat is a lot better than other feats and players feel "forced" into taking it, but its better than just giving away the feature by default. Maybe concentrating on multiple spells could also make all spells you cast against enemies roll with disadvantage for example, or maybe it could have some other balancing mechanic in play, but either way, I do feel if you are going to make changes like this, they should be carefully thought out to reduce the damage.
Alternatively, maybe it could work similarly to how martials gain extra attacks as they reach certain thresholds. Every 4 levels you can maybe concentrate on 1 additional spell, with an appropriate downside for doing so (something like all damage rolls always rolling for disadvantage while you are doing so), or a limitation like sum of the level of all spells you are concentrating on cannot exceed the maximum spell level you can cast (for example, a level 5 wizard would be able to concentrate on a level 1 and 2 spell or a level 1 and 1 spell, but not any combination which takes them over 3).
