Okay so I ordered one of the stamp kits that SeaWalnut suggested. It seems very nice, however it is too much for the metal label tape that I have, it kind of mangled it.
says "MORO" for Moro blood orange

Now, this tape is DYMO aluminum tape meant for the Rhino (the metal dymo embosser which is ~$200). I ordered it long ago thinking I could use it with any of their embossers, only to discover it is wider (not thicker) than the plastic tape and won't fit in the device. It feels really cheap and I believe is the same thickness as an aluminum can. In fact it is so thin I was thinking it would probably work fine in my cheapo plastic embosser if I cut it to fit. And it does! It got a bit wrinkled while I was feeding it in, and you have to tug on it very slightly to help it feed each letter.

So I wondered if instead of using the embosser I could just write on it with a pen. And that works fine also, using medium pressure.

My hole punch works fine on it also.
So I am thinking nobody should spend $200 on a dymo rhino, because you could just write on the same tape it uses with a pen, or find an aluminum tape that fits in the cheapo labelmakers, or cut it to size. Even with any of these, though, I wouldn't bother because the flimsy metal is almost certain to bend and tear with handling.
Using a thicker metal and the hammer/stamp kit is almost certainly the way to go for trees you really care about. I don't have any empty spray cans to cut up, but thicker metal sheep and nice pre-made metal tags are readily available.