Resolved
Overview of the issue
If you have more than 1000 taxonomy terms per taxonomy, your translations may not display correctly.
This is because translating taxonomy terms can be a very resource heavy task and can impact the performance. For this reason, WPML has built-in filter that limits the possible taxonomy terms per taxonomy to 1000.
Workaround
To change that limit, go to your wp-config.php file and add a code like this one:
define('WPML_TAXONOMY_TRANSLATION_MAX_TERMS_RESULTS_SET', '5000');
If you still have issues translating, increase the limit to a higher value.