As per this Wikipedia article many wines are apparently made using a catalyst that is an animal byproduct. The finings settle to the bottom and are removed. So when consuming the wine, there are no animal products in it. Would this be considered vegan?
The Vegan Society UK currently defines veganism as a lifestyle which seeks to avoid exploitation of animals and further clarifies that this means the avoidance of animal products and byproducts.
One thing all vegans have in common is a plant-based diet avoiding all animal foods such as meat (including fish, shellfish and insects), dairy, eggs and honey - as well as products like leather and any tested on animals.
Based on this definition, any wines that use animal parts such as isinglass (fish swim bladders) as catalysts or finings would not be considered suitable for vegans because the process causes unnecessary harm to animals. There are many wines that are made without using animal parts as ingredients or in processing, and Barnivore is a great resource for identifying those wines.
Many vegans will also avoid mead wine because it is created by fermenting honey with water and honey may be an ingredient of concern for vegans. It is interesting to note that early in the history of the Vegan Society the focus was exclusively on animal ingredients in food and didn't extend to concerns like leather or honey, so at the time mead would have been considered suitable for vegans.
Personally, I make a point of selecting vegan-suitable wines when I'm the one purchasing, but when somebody else is offering me a glass of wine I won't reject it based on the possibility that animal products may have been used in the process.
Veganism is not about what vegans eat. That is to say, what they eat is a major consequence of veganism, but it is not the primary concern. Instead, a vegan's goal is to try to eliminate unnecessary animal suffering caused by humans.
Given this, the reason that vegans don't eat animal products is because purchasing animal products economically supports animal suffering. Similarly, purchasing leather products economically supports animal suffering. And likewise, wine processed using animal products is not vegan, because purchasing it economically supports the purchase of those animal products, which in turn supports animal suffering.
I recently took a trip on British Airways; the wines on offer were labelled as either "suitable for vegetarians" or "suitable for vegans and vegetarians." I assume the former had been fined with egg whites or similar.
But clearly according to at least some authorities not all wine is suitable for vegans.
Would you eat bread sliced with a knife that just sliced a steak or cheese? This bread is no longer strictly vegan. I believe this to be the same principal as the wine you brought up, in essence this wine is NOT vegan, as some form of animal product has been used in the making of the wine even if this product is not an ingredient in the wine, this is similar to our sliced bread.
If no animal products are used in the making of this wine, then this wine would be considered vegan friendly.
Not sure if your wine is vegan friendly? There's a helpful online directory to help.
Edit: There are many reasons (such as ethics, health or religion) as to why someone chooses to have a vegan diet. My summary may not cover every ones personal reasons.