Python 3, 414 bytes

I hope that the order of the result doesn't count.

import re
t=input().replace("[", '(').replace("]", ')')
d={}
p,q="(\([^\(\)]*\))(\d*)","([A-Z][a-z]*)(\d*)"
for i in re.findall(q,t):t = t.replace(i[0]+i[1],i[0]*(1if i[1]==''else int(i[1])))
r=re.findall(p,t)
while len(r)>0:t=t.replace(r[0][0]+r[0][1],r[0][0][1:-1]*(1if r[0][1]==''else int(r[0][1])));r=re.findall(p,t)
for i in re.findall(q[:-5], t):d[i]=d[i]+1if i in d else 1
for i in d:print(i+': '+str(d[i]))

Python 3, 414 bytes

I hope that the order of the result doesn't count.

import re
t=input().replace("[", '(').replace("]", ')')
d={}
p,q="(\([^\(\)]*\))(\d*)","([A-Z][a-z]*)(\d*)"
for i in re.findall(q,t):t = t.replace(i[0]+i[1],i[0]*(1if i[1]==''else int(i[1])))
r=re.findall(p,t)
while len(r)>0:t=t.replace(r[0][0]+r[0][1],r[0][0][1:-1]*(1if r[0][1]==''else int(r[0][1])));r=re.findall(p,t)
for i in re.findall(q[:-5], t):d[i]=d[i]+1if i in d else 1
for i in d:print(i+': '+str(d[i]))

Try it online!

#Python 3, 414 bytes

I hope that the order of the result doesn't count.

import re
t=input().replace("[", '(').replace("]", ')')
d={}
p,q="(\([^\(\)]*\))(\d*)","([A-Z][a-z]*)(\d*)"
for i in re.findall(q,t):t = t.replace(i[0]+i[1],i[0]*(1if i[1]==''else int(i[1])))
r=re.findall(p,t)
while len(r)>0:t=t.replace(r[0][0]+r[0][1],r[0][0][1:-1]*(1if r[0][1]==''else int(r[0][1])));r=re.findall(p,t)
for i in re.findall(q[:-5], t):d[i]=d[i]+1if i in d else 1
for i in d:print(i+': '+str(d[i]))

Python 3, 414 bytes

I hope that the order of the result doesn't count.

import re
t=input().replace("[", '(').replace("]", ')')
d={}
p,q="(\([^\(\)]*\))(\d*)","([A-Z][a-z]*)(\d*)"
for i in re.findall(q,t):t = t.replace(i[0]+i[1],i[0]*(1if i[1]==''else int(i[1])))
r=re.findall(p,t)
while len(r)>0:t=t.replace(r[0][0]+r[0][1],r[0][0][1:-1]*(1if r[0][1]==''else int(r[0][1])));r=re.findall(p,t)
for i in re.findall(q[:-5], t):d[i]=d[i]+1if i in d else 1
for i in d:print(i+': '+str(d[i]))

#Python 3, 421 bytes

I hope that the order of the result doesn't count.

import re
t=input('> ').replace("[", '(').replace("]", ')')
d={}
p,q="\(([^\(\)]*)\)(\d)*","([A-Z][a-z]*)(\d*)"
for i in re.findall(q,t):t = t.replace(i[0]+i[1],i[0]*(1if i[1]==''else int(i[1])))
r=re.findall(p,t)
while len(r)>0:t=t.replace('('+r[0][0]+')'+r[0][1],r[0][0]*(1if r[0][1]==''else int(r[0][1])));r=re.findall(p,t)
for i in re.findall(q[:-5], t):d[i]=d[i]+1if i in d else 1
for i in d:print(i+': '+str(d[i]))

#Python 3, 414 bytes

I hope that the order of the result doesn't count.

import re
t=input().replace("[", '(').replace("]", ')')
d={}
p,q="(\([^\(\)]*\))(\d*)","([A-Z][a-z]*)(\d*)"
for i in re.findall(q,t):t = t.replace(i[0]+i[1],i[0]*(1if i[1]==''else int(i[1])))
r=re.findall(p,t)
while len(r)>0:t=t.replace(r[0][0]+r[0][1],r[0][0][1:-1]*(1if r[0][1]==''else int(r[0][1])));r=re.findall(p,t)
for i in re.findall(q[:-5], t):d[i]=d[i]+1if i in d else 1
for i in d:print(i+': '+str(d[i]))