What methods are available to evaluate similarity between different clustering algorithms?
If you just want to see how similar the clustering is between 2 algorithms, using the sklearn.metrics.adjusted_rand_score() function is a good starting point. This will work for unsupervised learning, no need for a label.

https://scikit-learn.org/stable/modules/generated/sklearn.metrics.adjusted_rand_score.html

Or are you looking at choosing the best combined grouping overall? I think "clustering the cluster" results is not a common approach, however there are a few articles online on how this could be done. But I don't know of any packages that will do this for you. 
One idea is to create a new column for each algorithm with the grouping chosen and then do some comparisions or calculations on these columns to get a final composite clustering.