Are curd and sour cream typical middle-eastern food?

This looks like it's most likely labneh, a form of yogurt that's strained to remove some of its moisture. The texture can range from a thick sour-cream consistency to something like dense cream cheese, and it's common in the cuisines of the Mediterranean and Levant. Sometimes it takes the form of rolled balls, sometimes it's served as a dip or sandwich ingredient, but it always has the characteristic sour taste of yogurt. Often it's garnished with mint or za'atar, but not always, though it's usually served with olive oil.

It's also referred to variously as "strained" or "Greek" yogurt (which can mean something different than the common slightly-thickened "Greek yogurt" that's currently trendy in the United States) so I wouldn't be surprised if there were a slight difference in translation that's causing the confusion. As you've found in your research, "sour cream" in the US usually denotes cream that's cultured in the same way as yogurt. "Curd" is more confusing; in the US it usually refers to cheese formed by treating with acid to produce solid clumps, but it can also refer to similar substances produced from fruit, as in lemon curd.

Is sour cream and curd different food, or the same? Is it really from middle-eastern cuisine, or just some restaurants that think it is, when it isn't? And if it is, what is the correct name of it?

Curds are the product of of milk that has been coagulated. It is basically the same process that makes your blood turn into scabs when we bleed. When acids like lemon juice or citric acid or rennet is introduced to the milk the process of curdling is initiated. This leads to the milk proteins (Casein) to form solid masses which is what results in curds.

Sour Cream on the other hand is cream that has been fermented with the use of certain lactic acid producing bacteria. The lactic acid that the bacteria produces thickens and sours the cream.