Both of these statements are correct and can be used interchangeably, and being available is not restricted to objects only. In fact, I had to fill out "availability times" for one of my jobs that told them when I could work during the week.
"Yes, I'm free."
"Yes, I'm available."
One of the definitions of available from the Merriam-Webster dictionary states: "free and able to do something at a particular time (should be available to meet next Tuesday)" Also, looking at synonyms of available, you can see that free comes up as a strong match, proving that they can both be used.
Other options you could use include:
"Yes, I do."
"Yes, I can."
Since the essential question was "Do you have 10 minutes to talk about a job interview" (emphasis mine), replying with "I do" means "I do have 10 minutes to talk..." in the context of the situation.
You can also use "I can" because you are essentially responding with "I can talk about the job interview."