Pg 118 PHB

Illusory Reality
By 14th level you have learned the secret... choose one inanimate, nonmagical object that is part of the illusion and make that object real. You can do this on your turn as a bonus action while the spell is ongoing. The object remains real for one minute.

RAW there's nothing against recasting Illusory Reality on the same object, if you did it would simply apply the Combining Magical Effects (Pg 205 PHB) rule which states:

Combining Magical Effects
The effects of different spells add together while the durations of those spells overlap. The effects of the same spell cast multiple times don't combine, however. Instead, the most potent effect -- such as the highest bonus -- from those castings applies while their durations overlap.

TL;DR: According to RAW, you can reapply this as a bonus action, and the duration would be extended, though the object wouldn't become more real.

The object is as real as it gets. It would be the same as creating any other object that can be seen and touched. The idea is to add some tangibility to your illusions via an additional aspect of magic (shadow magic). True Sight would reveal the illusion as false, but the object actually exists in space and so the creature sees that as normal. Whatever information the creature knows about the object is up to circumstances and your DM.

For instance, if the creature walked into an illusory house you made and you also made a chest real inside of it, there's no real way for it to know the chest isn't permanent (isn't real) unless he inspects it, or saw you create it, or otherwise reasoned its genuineness.

The fact that the object can deal no damage may or may not affect the way your DM rules the reality of the object. Surely a dagger made real should deal damage, and yet it cannot. My out-of-game reasoning for this is more about balance. I don't think the rule makers want you to use 1st-level spell slots to create grenades and daggers and other harmful things. In-game, i might suspect the material is weak or the object governed by special shadow magic rules. Your DM can handle the in-universe reasons for why a real object wouldn't be able to damage anyone.

So how do you handle a real weapon that can't deal damage with true sight? I think there are a number of explanations you could have for why a real dagger wouldn't deal damage, but at the end of the day, all that matters is that True Sight sees the object as real, because it's not an illusion. See is the key word here, since true sight has nothing to do with inspection, touch, or investigation. True Sight is set apart from intelligence. Think of it like a plastic bowl of fruit. It's real, right? But, until you actually inspect the fruit, you don't recognize it as a fake real item. Real being opposed to illusory here, not opposed to genuineness.