Java (JDK), 110 ¹²² bytes

-12 bytes after the added possibility to output the list as a string joined with linebreaks

Port of my JS answer, see explanations here

s->{var r="";int a=0;for(var c:s.toCharArray())r+=c==','&(a+=("{[()]}".indexOf(c)+4)%7-3)==0?"\n":c;return r;}

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Java (JDK), 107 ^{110 122} bytes

-12 bytes after the added possibility to output the list as a string joined with linebreaks
-3 bytes thanks to ceilingcat

Port of my JS answer, see explanations here

s->{var r="";int a=0;for(var c:s.toCharArray())r+=c==44&(a+=("{[()]}".indexOf(c)+4)%7-3)==0?10:c;return r;}

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Java (JDK), 122 bytes

Port of my JS answer, see explanations here.

s->{var r="";int a=0;for(var c:s.toCharArray())r+=c==','&(a+=("{[()]}".indexOf(c)+4)%7-3)==0?",,":c;return r.split(",,");}

Try it online!

Java (JDK), 110 ¹²² bytes

-12 bytes after the added possibility to output the list as a string joined with linebreaks

Port of my JS answer, see explanations here

s->{var r="";int a=0;for(var c:s.toCharArray())r+=c==','&(a+=("{[()]}".indexOf(c)+4)%7-3)==0?"\n":c;return r;}

Try it online!