I went perusing through the Magic Item listings, and I found the Animated Shield:

While holding this shield, you can speak its command word as a bonus action to cause it to animate. The shield leaps into the air and hovers in your space to protect you as if you were wielding it, leaving your hands free. The shield remains animated for 1 minute, until you use a bonus action to end this effect, or until you are incapacitated or die, at which point the shield falls to the ground or into your hand if you have one free.

The obvious thing here is that it frees up your off-hand, allowing you to use a 2 handed weapon while still benefiting from the use of a Shield.  But how does that interact with the Cleric/Paladin's ability to use a Shield as a spellcasting focus?  Since the specific phrase "as if you were wielding it" is present, that would suggest you are allowed to use it as if you were, in fact, holding it.  But that is likely referring to the AC bonus and the defensive bonus for the handful of AoE spells that mention it.

Would a Cleric or Paladin be able to use an "activated" Animated Shield as a spellcasting focus?

I believe that is one of the intended purpose of the Animated Shield. If the shield did not function in exactly the same way as when inanimate, while allowing you to use your now free hand in whatever manner you choose, what would be the purpose of the enchantment at all?