I have a piece of software I wrote for work that captures stderr to a file like so:
import sys
sys.stderr = open('C:\\err.txt', 'w')
so it’s definitely possible.
I believe your problem is that you are creating two instances of writer.
Maybe something more like:
import sys
class writer(object):
log = []
def write(self, data):
self.log.append(data)
logger = writer()
sys.stdout = logger
sys.stderr = logger