Some improvised weapons are 'items-at-hand', never intended to be weapons, such as the list from the rules given in Mars Plastic's answer: "broken glass, a table leg, a frying pan, a wagon wheel, or a dead goblin."
However, other objects are intended to be used as weapons, that is, designed to damage creatures, and still count as 'improvised'. Things like Holy Water, Thrown Oil, and Alchemist's Fire are "treated as an improvised weapon". The Weapon Master Feat does not allow one to gain proficiency in throwing oil flasks, for example.
Natural weapons are the parts of a creature's natural form which are made to be used as weapons, "such as a claw or tail spike". They are the attack mode accessible to a druid in just about any of their wild shapes. Note that wild shape permits the druid to "retain the benefit of any features from your class, race, or other source and can use them if the new form is physically capable of doing so." Here, 'other source' would include Feats, so if the Weapon Master feat allowed proficiency with natural weapons, it would permit a druid with the feat to gain proficiency (and use their own proficiency bonus) with specific natural weapons, such as claw and bite, that they might use while wild-shaped. However, by limiting the choices to simple and martial weapons, this is not an option for the druid.
Neither would the Weapon Master feat permit a PC race with natural weapons (such as Aarokokra, Centaurs, Leonin, Lizardfolk, Minotaurs, Satyrs, Tabaxi, Tortles) to gain proficiency in their use.
Besides literally having 'weapon' in their name (they could, after all, have been called siege engines), we know that siege weapons count as a type of weapon from the description of Daern's Instant Fortress, which says (DMG 161):
The roof, the door, and the walls each have 100 hit points, immunity to
damage from nonmagical weapons excluding siege weapons, and resistance to all other damage.
What is not clear, however, is who makes the attack roll for a siege weapon - the weapon itself as an object, or one of the crew, albeit without a proficiency bonus. In a game where one of the crew makes the roll, Weapon Master cannot grant proficiency with a siege weapon to allow a crewmember to add their proficiency bonus to the siege weapon's attack roll.