There is nothing in the rules that I know of that says an object must be used in only one manner in any given turn - and the only rule that mentions it (Cleric using a shield with a holy symbol on it) says you can use it as both a shield and a holy symbol simultaneously, which is precedent for what we're discussing.
Let's start that the rules do not show every single interaction of every rule with another, so we won't find either of our points spelled out in black and white. I think that there is reasonable amount of guidance in the rules but I don't know if it will overcome a significant enough burden of evidence to convince everyone.
Using Player Basic Rules v2.0 (standard, not the printer-friendly version)
(pg 44)
Shields. A shield is made from wood or metal and is carried in one hand. Wielding a shield increases your Armor Class by 2. You can benefit from only one shield at a time.
(pg 47)
Improvised Weapons
Sometimes characters don’t have their weapons and have to attack with whatever is close 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.
Key word in both of these to me is "wielding". And the question seems to be if you can wield something as a shield at the same time you are wielding the same thing as a weapon.
In the basic rules I searched on "wield" and got no formal definition of it. If you can find something in the actual books that might shed more light. Otherwise I'm going to have to look farther afield.
Weapon Proficiency and Shield Proficiency are separate proficiencies. You do not automatically have one if you have the other. So they are not done the same way because that would require them being both granted together if it was only one thing.
So I have this object. I can Wield it using my Shield Proficiency. It then gives me +2 AC. I can Wield it using my Weapon Proficiency (or lack thereof, but improvised weapons still need to be wielded to be used) for various effect.
And this is where I think we disagree. I see the rules requiring you to wield them in different ways because it takes different proficiencies. What am I missing in the rules (besides "silence") that counteracts that?
Fine, a set of thieves tools. A rogue Thief of the appropriate level can use their thieves tools as a bonus action (which involves their proficiency) and then attack you with those same thieves tools as a regular action using them as an improvised weapon.
Tool Proficiencies are also not Shield Proficiency. At my table you could not use tools to pick an lock and attack with the same tools in the same turn, or play a violin while also attacking with it. (Unless you have the Feat: Lindsey Sterling)
But that follows the same logic I used earlier, because it's not clearly spelled out either way. If it works for your table, go for it - that's really what matters.