Interesting idea! That could be implemented without much difficulty.

Would you prefer to have it as a status text (green if you've already had it and red otherwise), speech text from the character, or something else entirely?

Not the OP but a speech text from the character sounds more immersive and fun. Like when you talk to yourself while working as many people do IRL.

kobakumadaze has good idea. Whatever you implement im sure it will be fine. Right now i just open the menu, pause the time and search stuff in my home inventory. Its not that troublesome, but i feel checking by right clicking could be more organic to the game flow.

Well, @KoBakumaDAZE, I'm fine either way, but since you're the first one to respond, I'll let u have the first slice of cake. ;3

Fastest way would be to have an icon or as you said text color highlighing the items already in posession. KoBakumaDaze way is immersive, but could be optional as toggle, while the most efficient way would be default.

+1 to the idea of checking via context menu. Maybe for simplicity it could just automate opening the existing Home Inventory panel via the context menu, with the search field already filled. That way it would show the amount you have and where it's stored.

Maybe not ideal for some generic names where one item name could match multiple others, but good enough to speed up a lot of the manual re-typing.

Something like how it works in these mods:
What Can I Craft Addon
[B42] Neat Crafting V1.5

For everyone in this discussion, as of 0.5.0, the number of a specific item in your home inventory can now be seen directly in an item's tooltip when it is hovered. Hopefully, this will save you guys a right-click each time you want to check.