By strict RAW, no. Allowing it wouldn't harm your game meaningfully, however.
Shields are in the armour category, and are thus not weapons. You could easily argue that you could use it as an improvised weapon, but that does not make it an actual weapon. Using improvised weapons as justification for doing this does not hold up, as that would allow you to weapon bond to anything you could feasibly swing around. Including a goblin, if you're strong enough.
The biggest impact allowing this would have on your game, however, is the fact that putting on a shield normally takes an action, while allowing weapon bond allows you to do the same as a bonus action. This is not a massive benefit, and many DMs would allow you to have the shield in hand at the start of combat anyway.
Furthermore, Weapon Bond is what is referred to by the game designers as a "ribbon" ability. Ribbons are abilities that, while flavourful, don't have a tangible effect on game balance.
In short, by a strict reading of RAW, it's not allowed. However, I have not come across a single DM so far that wouldn't allow an Eldritch Knight to weapon bond to his shield, and it will not break your game balance.
This depends on DM ruling.
The rules don't explicitly cover this case, so there are two, equally-valid ways to rule this.
The first is that improvised weapons don't count as weapons. This ruling is supported mainly by a few parts of the Improvised Weapons description (page 147, PHB). It says:
Sometimes characters don't have their weapons and
have to attack with whatever is dose at hand... In many cases, an improvised weapon is similar to an actual weapon...
These line heavily imply that, while improvised weapons often act like weapons, they actually aren't. Since magic weapon specifies that it can only target weapons, by this ruling improvised weapons can't be made magic.
This interpretation is supported by Jeremy Crawford, who tweets:
Weapon Bond works with a bona fide weapon ("Behold, my sword!"), not an improvised weapon ("Look, a stool!").
The second interpretation is that, since the phrase "improvised weapons" contains the word "weapon", then improvised weapons must be weapons. With this interpretation, anything that can be done to a weapon can also be done with an improvised weapon.
Both of these interpretations are valid, and based on valid rules reasoning, so this is going to be dependent on your DM.
Best Answer
This is one of those situations that the rules don't really cover and as such would be left up to the decision of the DM.
Personally, I'd allow it for the sheer amusement of the player(s) but it would very much depend on what exactly is being used as the improvised weapon.
From the Improvised Weapons section on page 147 of the PHB:
By this statement a DM could definitely treat objects similar to actual weapons as weapons for the purpose of a weapon bond.
However, the Improvised Weapons section also says:
By this statement the players could find a lot of things that wouldn't fall within the "similar to actual weapons" clause and allowing "any object you can wield with one or two hands" to be subject to the rules of the weapon bond could create some pretty hairy situations.
As such I would allow improvised weapons to be bonded but would restrict it to objects that are similar to actual weapons and "common place" objects.
Also, this isn't defined within the rules but I see improvised weapons as just that, improvised, makeshift, and temporary. They are objects that wouldn't normally be classified as weapons except for under certain circumstances. As such it would be reasonable to say that Weapon Bond wouldn't work with improvised weapons because they aren't normally classified as weapons.