The Search API module shows a two-words taxonomy term as two separated words, in its widget I am using taxonomy terms to classify my content, and the Search API and Facets modules for searching.

I have some nodes that are tagged with "Asian Cuisine" term; after indexing the content, "Asian Cuisine" is handled as two separated words. 



Is there a way to avoid it happens?
Based on your screenshots, consider using the "Taxonomy Term" type for the field indexes. This should remedy your issue.

By using Full Text, you're breaking up the taxonomy term names into searchable words. Use either the default taxonomy term value, or try using the "All Parent Terms" field.