Hi!
Depends on where you are located and local customs regulations. In Germany (as recipient/addressee) gifts
can be subject to 13% fixed rate (duty) instead of the otherwise applicable rate based on what product it is. Let's say you get a "goodie bag" from your friend for Christmas that exceeds the applicable threshold of 150 Euro for import duty: A T-Shirt, a bottle of wine, a coffee table book, some (importable) snacks/foodstuff ... Instead of finding out what the T-Shirt costs, adding the approx. 29% rate, and so on. They
can simply go with the 13% over the total amount + VAT. In some cases this will be significantly cheaper.
The problem is proving it (getting customs to believe you

), especially with pricey merchandise. This is really meant for close relatives/family sending you a real gift.
With an invoice in your name - this is born to fail. They treat it the same way as if you would have declared 100$ on a 1000$ item. They ask for the correct amount/invoice and proceed by treating it as the merchandise it is. Some try to go the "I don't have a invoice." route, which then leads to them researching the value. So if you bought this at a better price than what customs will find as MSRP, it's in your best interest to provide that invoice.

(Speaking solely for Germany!)
In the end, declaring it a gift, while it isn't/wasn't one - same situation as with under or not declaring. It will absolutely do no good in the process of shipping an item that will result in an anticipated four figure import fee.
I'm sorry, but over the years I've seen and heard it all - and that's what it comes down to. In the very early days of my intl. shipping I had established a good business relationship with an officer at my local customs office. There were a lot of things that we managed to "square away" by me being honest/ proving correct paperwork to my best ability, and the officer not nit picking to any painful extent. So this meant that if out of the many parcels I had, one would have "****ty paperwork" or - because of the relationships build abroad through the business I conducted - was a gift, she wouldn't make it any harder on me than absolutely necessary and/or give me a pass on it - based on my verbally declared (correct!) amount. She loved Oreos, so from time to time, my shipments from the US would contain a package or two, or new flavors/styles that weren't available in Germany. And they would magically stay at the customs office break room. In other words, it was a human experience and we knew each other by name. And just for the record, nothing unlawful ever happened.
With the raise of intl. shipping this is pretty much unthinkable these days - besides another customs office being responsible for my area now. And there, the officers go by the "full extent of the book" down to the last dot - any and every time. And I've witnessed the people the recipients trying: "it's a gift", "it's only worth 30 Euro", "but it's a genuine product" and other lengthy arguments that lead to nothing for the recipient. ... But I also witnessed customs officers nit picking to a painful extent, when merchandise was destroyed while inspecting it without any probable cause ...
That's why I simply rely on couriers now, for the few shipments I deal with these days, that deal with customs on my behalf and while the parcel is en route. No more headaches for me.
Kind regards,
Oliver