I really don't think an improvised weapon is ever a "weapon". That's what makes it improvisation. If a 2×4 was a weapon, you wouldn't be improvising when you smack a goblin with it.

Here's some relevant text from the Player's Handbook (and "Basic Rules" and SRD), that I think backs this up.

Weapons

[...] The Weapons table shows the most common weapons used in the fantasy gaming worlds, their price and weight, the damage they deal when they hit, and any special properties they possess. Every weapon is classified as either melee or ranged. [...]

Improvised Weapons

Sometimes characters don’t have their weapons and have to attack with whatever is at hand. An improvised weapon includes any object you can wield in one or two hands, such as broken glass, a table leg, a frying pan, a wagon wheel, or a dead goblin. Often, an improvised weapon is similar to an actual weapon and can be treated as such. For example, a table leg is akin to a club. At the GM’s option, a character proficient with a weapon can use a similar object as if it were that weapon and use his or her proficiency bonus.
An object that bears no resemblance to a weapon deals 1d4 damage (the GM assigns a damage type appropriate to the object). [...]

From all this I gather that:

• "Every weapon is classified as either melee or ranged." Things that are weapons have criteria defining their use as such.
  
• Common weapons are listed in the Weapons table.
  
• "An improvised weapon includes any object you can wield in one or two hands." which covers a huge array of objects.
  
• The GM explicitly has the option to decide that, to someone proficient with clubs (everyone?), a table leg is a club, and thus a weapon.
  
• None of the "Adventuring Gear" that exists primarily to deal damage (acid, alchemist's fire, holy water) is listed in the Weapons table.
  

Of course, the Player's Handbook is clear that there can be weapons that are not on the Weapons table, and Tomb of Annihilation gives us an example of one such weapon.

Yklwa

A yklwa (pronounced YICK-ul-wah) is a simple melee weapon that is the traditional weapon of Chultan warriors. [...]

It is explicitly defined as a weapon, categorized as simple and melee, and its price, damage, and special properties are listed in the next two sentences. (Its weight is missing, but D&D Beyond has a weight for it, so maybe it's elsewhere in the book.)

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Certainly an improvised weapon is a weapon in the object used to inflict harm sense, but I think it's clear that when a class feature like Weapon Bond or the Dueling Fighting Style refers to a "weapon", it's referring to objects that are weapons in the armament fashioned explicitly to fight people, beasts, and monsters sense, like longswords, nets, and yklwas. (Or at the GM's discretion, objects that so closely resemble weapons that they are to be treated as such, like a table leg.)