Apparently.
Like; I can hit you with a table leg, a carpentry happer, or an inanimate carbon rod... but If I do so, I cant get the bonus for dueling, or smite, or rage.
But if I instead use a club or a mace, I'm suddenly golden again.
And I can wield a mace, nunchucks, tonfa, fighting stick or a club in each hand, but not a carpentry hammer, a pool cue snapped in half, steel wrench or an iron bar in each hand.
Dont mention this to Steven Segal or Jackie Chan by the way. I think we just invalidated half their films.
All the examples you have are close enough to known weapons to be considered weapons. What about a table, a chair, a cart's wheel, a wooden box or even the body of a dead opponent?
It clearly stated that if an improvised weapons looks like a defined weapons then it will behave as that weapons. A shield does not bear any resemblance to any weapons so it does not qualify for a weapon in respect to combat manoeuver requirement.
As stated in phb, the definition of a weapon in the phb is quite clear and your weapon must fall in one of the two categories; that is either simple or martial. A shield, unfortunately, does not qualify.
As for the feats they all specifies weapons, yes, but from the game's perspective. Not from the dictionnary one. So again, the shield does not qualify.
For the fighting styles, look at it from à single class perspective. Only the champion ever gets two. Multiclass is supposed to be an option. So without mc you only get one style. Is it the RAI that you should only use one style at a time? I do believe so. Can you know more than one style? Sure you can and that is a great thing. That enables you to adapt much more to different combat situations. Juste like many martial artists practice different styles like jujitsu, kung-fu, aïkido, karate, and judo to name but a few. Do they use more than one style? Of course they do. But not at the same time. They choose from moment to moment which style is the best at that moment and use it. In game turn, it means that they change style from round to round.
The power of choosing between styles is already strong enough. Please do not stack them. Neither of our position is quite as RAW as I would like, but I do believe that mine is as RAI as possible. I might be wrong, I was not on the design team.