That's the right definition. I.e. variable costs are a cost that increases when 1 more unit of a product/service is sold.
The answer to your question truly depends on the cost structure of your company.
Does your company employ x call centre staff with an annual salary of $xxxk per year? If so, Fixed Costs.
Alternatively, does your company pay call centre staff hourly, and have them go overtime when there are more complaints? And/or, do you outsource the calls to, say, the Philippines, and pay per call/minute? In which case, we're talking Variable Costs.
All that being said, I would generally assume customer care/service is Fixed Cost. Why? It's included in the SG&A AND any changes are step changes. What I mean, is, if you sell 1,000 more goods don't you have to build another factory and therefore factories are variable costs? Of course not...these are larger investment decisions :). Likewise, increasing call centre capacity is generally a step change decision (i.e. buy another call centre, hire 100 more people, etc. etc.)