What universe have you been living in that this comes as a surprise to you?

If someone uses religious beliefs to justify subjugating women, would they not be sexist?

If someone uses religious beliefs to oppose interracial marriage, or evince disgust and interracial sex, would they not be racist?

The cause of phobia can be anything. External observers really cannot say. But if you are willing to state in a public forum that homsexual acts, women doing work, or the mixing of races disgust them, then yes, they are being homophobic/sexist/racist.

Let me give you an analogy. Communist China under Mao criminalized homosexuality, and enforced those laws quite stringently. If someone who genuinely believed in that Communist ideology said that they believed homosexuals belonged in jail, not because of homophobia, but because they supported the rule of law, would you give them a pass from their homophobia?

Or lets say someone in Saudi Arabia said, last year, that they weren't sexist for believing women who drive are disgusting. They believe it is a sin, per their religion. Would you say they weren't sexist for, after all, simply believing in their religious tenets?