Indeed, it does, particularly if you construe culture as something that not only refers to countries, but also sub-areas within countries. As an example, you'd imagine that customers in China would view customer service issues differently than people in Canada. What we don't tend to focus on is that culture works much more "locally". So, for example, what constitutes good customer service is New York City is different than what passes for customer service in rural Nebraska.
Important Differences Obscured By Generalities
Many of the things talked about as engendering good customer service are so general that they obscure how different people are who come from different settings -- urban vs. rural, small town vs. large city, sub-culture vs mainstream culture (i.e. Hispanics vs Anglo Saxon). When we talk generalities, let's say about respecting the customer, or treating customers as you would yourself like to be treated, you'll entirely miss the fact that what YOU think constitutes respectful behavior will often be different from what a customer with a different background and set of values. We ALL want respect. What respect means will differ between say, a medical doctor or professor, and a person from an inner city environment.
In fact, generalities about how to treat customers, and what customers want actually obscure the fact that when it comes to behaviors, culture becomes critical.
An Example: Big Vs. Small, Fast vs. Slow, Friendly vs. Efficient, Montreal vs. Winnipeg
Having lived in two quite different cities, Montreal and Winnipeg, I've had the chance to experience first hand how culture affects the customer service process. Note: If you aren't familiar with Canada, think NY City and Fargo, North Dakota instead) Montreal is a cosmopolitan, large, fast paced city, influences by a Quebecois culture unique in Canada. Communication in Montreal is very open, louder, emotionally expressive. Winnipeg, a prairie city, is much smaller, slow paces, and friendly in a chatty way. Montreal is friendly, because people are so open to each other, and interested in each other. Winnipeg is friendly in a different, more reserved way. Cultures are different.
If you spend time in Montreal, but were born in Winnipeg, you'd find Montrealers rude, uninterested, curt, and you'd feel that you often couldn't get the time of day from people working in some of the establishments there. You might feel that employees there feel it's YOUR priviledge to be served.
If you spend time in Winnipeg, but were born in Montreal, you'd be frustrated at how long everything takes. You can wait in line at a convenience store even when there are no other customers. Employees chat amongst themselves keeping you waiting. You might consider them lazy. It's agonizing.
The reality is that these differences are cultural, so the behaviors don't mean the same thing to a native Montrealer or a native Winnipeger. It's not that either is more or less customer oriented, or is actually delivering better or worse service WITHIN THE CONTEXT OF THE CULTURE. Customers in Montreal want fast, not chatty. Customers in Winnipeg expect more chatty friendliness. What constitutes rudeness to the Winnipeger in Montreal is actually efficiency -- doing the job for Montrealers.
For more on why respect and common sense don't work in customer service see: Isn't Customer Service About "Respect" and Common Sense?