By the strictest definition, the Caspian Sea is a lake as it is surrounded by land. A more proper definition would probably be endorheic basin as water from the Caspian Sea never flows to the ocean, similar to the Great Salt Lake,the Dead Sea, and what is left of the Aral Sea.
However, the Caspian Sea displays several traits which distinguish it from what is normally considered a lake. The most obvious difference is its enormous size. The surface area of the Caspian Sea is 371,000 square kilometers, larger than Germany and four and half times larger than the next largest lake, Lake Superior.
The Caspian Sea contains brackish water, with an average salinity about halfway in between freshwater and ocean water. The salinity varies greatly though, with the northern part being considered fresh while a section called the Garabogazköl has salinity sometimes ten times greater than the ocean. The average salinity of the Caspian Sea is roughly similar to the Baltic Sea and the Black Sea. Outside of the northern part, it is too salty for use as drinking water or for agricultural use.
Given its size and salinity, early settlers probably didn't give much thought to the question of whether or not to call it a lake or a sea. As far as they could tell, it was a sea, and the name stuck.
There is also a bit of a linguistic difference happening. Nearly all lakes which are referred to as seas are in Central Asia and the Middle East. This indicates that perhaps the languages in these regions define lakes and seas differently. Whereas in English, it's whether or not it's surrounded by land, other languages may consider salinity the distinguishing characteristic. For agricultural societies, this distinction is much more useful as the name tells you if the water can be used for farming. For a maritime society like Great Britain was, knowing which bodies of water will connect you to the ocean is more important.