People can interpret it however they wish, but this is not a "religious belief" matter. A matter on religious belief must include "undue hardship" on a business. The clerk who refused to issue gay marriage licenses was overruled by every single court, even though she could decline to sign them. That is not an overturning of a law. And I read the language used in the brief.
That article story referenced a SB 180, a bill introduced into the Kentucky state senate on 18 Feb 2016, and the mention of "allowing discrimination against interracial couples" led some readers to infer that the bill in question had made (or would make) mixed-race marriage illegal in Kentucky.
The bill made no mention of interracial couples, however. Instead, it sought to clarify that under the provisions of the Religious Freedom Restoration Act enacted by Kentucky in 2013, businesses could not be punished for violating local ordinances that prohibit discrimination based on sexual preference or gender identity if the discriminatory practices were based on "sincerely held religious beliefs":
Government shall not substantially burden a person's freedom of religion. The right to act or refuse to act in a manner motivated by a sincerely held religious belief may not be substantially burdened unless the government proves by clear and convincing evidence that it has a compelling governmental interest in infringing the specific act or refusal to act and has used the least restrictive means to further that interest. A "burden" shall include indirect burdens such as withholding benefits, assessing penalties, or an exclusion from programs or access to facilities.
The matter of inter-ethnic marriage and its constitutionality was already eliminated on the grounds of unconstitutionality by The Supreme Court in 1967 ending the Racial Integrity Act of 1924. So this is not germane to the matter at hand.
It was just a story that was circulated. None of that language was actually included in SB 180. But people sometimes imagine things without use critical thinking to verify a rumor.