No. The player is a living entity, but not a mob.
From a terminology standpoint, the word "mob" usually refers to hostile, non-player creatures.
In Minecraft's source code (courtesy of MCP), we can see for ourselves how the game classifies different entities. The player class
EntityLivingBase, whereas all hostile creatures extend
EntityMob. The player is not considered a mob in the game code either.
However, in many cases, things that apply to mobs apply to all living creatures. When you see that something deals x damage to mobs, that really means it deals x damage to living creatures. A diamond sword deals as much damage to a skeleton as it does an unarmored player.