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.