Chinese Proficiency tests available
The one for the PRC (mainland China, HK, Macao) is the 汉语水平考试, abbreviated to HSK, directed at non-native speakers (including overseas Chinese heritage learners as well as foreign students and workers). The test is administered by Hanban, part of the Minstry of Education of the PRC.

The equivalent for the ROC (Taiwan and its territories) is the 華語文能力測驗, abbreviated to TOCFL. It is also used as a qualifying credential for the Taiwan scholarship, applying for universities and job specifications. It is administered by the SP-TOP, under the direction of the Ministry of Education of the ROC

There is also a 职业汉语能力测试 (ZHC) for native speakers in mainland China, which is under the Ministry of Labor and Social Security, and reserved for those PRC citizens who want to go into certain fields e.g. journalism, judiciary, translation.

The proficiency test for PRC citizens hoping to work in broadcasting, education or the government is the 普通话水平测试, with its very clear emphasis on Standard Mandarin as strictly defined by the latest standard. 

There is also a HK-specific English and Mandarin proficiency test for teachers there: 教師語文能力評核 (LPAT).

These are all Mandarin tests under their respective standard of Mandarin; I have not heard of such a government (as opposed to a university or medical) test for Hokkien, Hakka, or even Cantonese.