toutput = {}
for entry in input_dictionary:
    if entry['name'] not in toutput:
        toutput[entry['name']] = []
for p in entry['playing']:
    if p not in toutput[entry['name']]:
        toutput[entry['name']].append(p)
output = list({'name': n, 'playing': l} for n, l in list(toutput.items()))
Объединение списка словарей в python