#Python 3, 196 170 160 154 bytes

Awkwardly long, thanks to Mego for saving 6 bytes:

d=y=""
for C in input():
 for a in "[](){}":y+=C*(C==a)
 y=y.replace("()",d)
x=y
for r in y:x=x.replace("{}",d)
for s in y:x=x.replace("[]",d)
print(x==d)

Python 3, 196 170 160 154 bytes

Awkwardly long, thanks to Mego for saving 6 bytes:

d=y=""
for C in input():
 for a in "[](){}":y+=C*(C==a)
 y=y.replace("()",d)
x=y
for r in y:x=x.replace("{}",d)
for s in y:x=x.replace("[]",d)
print(x==d)

#Python 3, 196 170 160 bytes

Awkwardly long, still golfing:

d=y=""
for C in input():
 for a in "[](){}":
  if C==a:y+=C
 y=y.replace("()",d)
x=y
for r in y:x=x.replace("{}",d)
for s in y:x=x.replace("[]",d)
print(x==d)

#Python 3, 196 170 160 154 bytes

Awkwardly long, thanks to Mego for saving 6 bytes:

d=y=""
for C in input():
 for a in "[](){}":y+=C*(C==a)
 y=y.replace("()",d)
x=y
for r in y:x=x.replace("{}",d)
for s in y:x=x.replace("[]",d)
print(x==d)

#Python 3, 196 170 160 bytes

Awkwardly long, still golfing:

y=""
for C in input():
 for a in "[](){}":
  if C==a:y+=C
 y=y.replace("()","")
x=y
for r in y:x=x.replace("{}","")
for s in y:x=x.replace("[]","")
print(x=="")

#Python 3, 196 170 160 bytes

Awkwardly long, still golfing:

d=y=""
for C in input():
 for a in "[](){}":
  if C==a:y+=C
 y=y.replace("()",d)
x=y
for r in y:x=x.replace("{}",d)
for s in y:x=x.replace("[]",d)
print(x==d)