Tag Groups was developed to work with the WPML plugin. Since the Polylang plugin tries to emulate some of the WPML functions, users have reported that Tag Groups also works well with Polylang.
Although we try to ensure also the compatibility with Polylang, we are currently only able to ensure the compatibility with WPML.
You can test most of the features that are relevant for translating your website with the free Tag Groups plugin.