Is there a Chinese equivalent of the Japanese Kanji Kentei? There is a test called Nihon Kanji Nōryoku Kentei or 日本漢字能力検定 , that tests Japanese kanji (characters that came from Chinese). 

It was originally meant for native speakers of Japanese, and it measures the ability to correctly use kanji, their readings, meanings, stroke order, four-character idioms etc. Korea also has a quite similar test for their Hanja called 全國漢字能力檢定試驗. 

These tests should not be confused with other certifications like JLPT (日本語能力試験) or TOPIK (韓國語能力試驗) that are indeed meant for foreigners and that are comparable, I guess, to the HSK.

Do we have any similar certifications in Chinese (that tests primarily Hanzi, their correct usage and pronunciation and were intended for native speakers)?
There is a similar test in mainland of China. But only targeting on people who looking for the specific job position.

It's called "汉字应用水平测试". 
http://www.sppc.edu.cn/2e/8a/c66a11914/page.htm

And this is an official test. It's managed by ministry of education of the PRC.
http://www.moe.gov.cn/s78/A18/yys_left/moe_803/