I try. But it's easy to forget on a day-to-day basis that those people you're working with every day (and eating lunch with, and gossiping with, and going to happy-hour with after work) are actually your Customers. Until you're the one who's been "wronged"... Like I said, it's easy to forget about the customer service function when you're among a gathering of friends.
As for whether the customer is always right, the answer is "It depends." (Consultant's Bible, Rule #1: The most appropriate answer to any question is "It Depends".) I don't believe that the customer is ALWAYS right, but then again, in Service, "Who is Right?" isn't the question. If you tell the customer they're right when they're not, you may be teaching them bad habits that hurt both of us in the long run. A true customer service professional will educate their customers on the "right" way of doing business, while still keeping them happy. Sometimes this means saying "You're right, and I'll make sure our mistake doesn't happen again". Sometimes you have to say "I'm sorry if I wasn't clear when I explained to you what we need from you -- here's a credit to compensate you for your trouble, and would you please do it this way in the future?" Sometimes you have to say "I understand your concern, I'll do what I can to address it, meanwhile, here's how you might handle this situation in the future".
The customer isn't always right, and a service-provider doesn't really have to pretend they are. However, a service provider DOES have to remember who pays the bills, and do everything they can to keep the customer happy.
As for the Employer... I'm a consultant, and I work for placement companies. Which often means that *I* am the customer of my employer. But I have to keep in mind that my employer's OTHER customer is the Client, and THEY are the ones we have to keep happy. The client is my customer as well, but at the same time, the project manager that I work for has to give me what I need to get my job done. So I am THEIR customer at the same time that they are MY customer. It's a triangle that makes for some interesting dynamics...