How would one confirm that the ashes in the urn are from who their supposed to be?

In Ontario, Canada, the regulations are that the crematorium places a metallic ID tag with the body. That tag follows the body throughout the process and will be included with the ashes. The urn should come with some paperwork saying "cremated remains of John Smith #2875" and the urn should have a metal tag with "2875" stamped on it inside. I'd assume other places follow similar regulations.

This type of system prevents mistakes, but it doesn't prevent the crematorium staff from lying and producing fraudulent tags and paperwork. At some point, you just have to trust the cremated remains you've been given are the right ones and the staff aren't purposefully lying to you.

Just make sure you aren't cremated in Eastern Europe
I watched a documentary a while ago about how crematoriums in Eastern Europe, especially in war torn areas like Ukraine, routinely collect bodies and then spend the time cutting up the bodies while in their care for organ harvesting. One of the biggest businesses out there apparently is bone harvesting .... bone material that is used for dentistry, bone repair and all kinds of things. Bone grafts and powder that makes its way to the medical market in first world countries like the US and Canada.
It made me think of the process of having a tooth pulled ... they often use bone grafts to help the healing process of a newly extracted tooth. That bone graft material had to come from somewhere.
Once the body is in the care of a private business .. who knows what they do with it. Then the body is cremated and no one would ever know that the body had been disrupted and cut up.
I don't think stuff like that happens in Canada or US ... but we do just blindly put our full trust in these business to collect, store and process dead bodies for us without much oversight or monitoring. Part of the documentary did mention that in the early 2000s, there was one morgue in New York state that had someone dissect bodies to collect parts illegally before they were found out. After that, they had to tighten up laws and security and regulations dealing with the dead.
I felt weird just mentioning all this so I went to look up the documentary .... here it is ...
Body Snatchers Inc. : What If Your Implants Were Stolen Organs?
https://youtu.be/pENORq4KlwU
- I don't think stuff like that happens in Canada or US
  sigh. I can't speak for Canada, but unfortunately it does happen in the US: California, 1980's. Colorado, 2010's. And the one that brought it to my attention, New Jersey, 2006:
  The indictment was the first set of charges to come out of a widening scandal involving scores of funeral homes [that sent the bodies for cremation] and hundreds of bodies, including that of “Masterpiece Theatre” host Alistair Cooke, who died in 2004. The investigation has raised fears that some of the body parts could spread disease to transplant recipients.
- Apart from bones, what can you take from a dead body? Organs have to be "alive" to be taken, so you can only harvest organs from brain dead people and still under life assistance
- I find this really hard to believe.
  I'm sure that unauthorised organ harvesting has occurred in isolated circumstances.
  But I'm incredulous that it could happen on an industrial scale.
- I had bone graft put in after I got my wisdom teeth out but I'm pretty sure my dentist told me it was some sort of animal bone, not human. I'll trust him cause that's slightly less creepy to me.

A family member that had screws in their leg was cremated. The screws were in the can of ashes. Pretty sure it was them.

Where i work we need to remove all the metal from the ashes. Because they need to pass-through a grinder and metal would damage it. The only real identication is from the litle fireproof stone with a number, and our internal log keeping which we do a lot of.

There's a metal identification tag that goes with the body the entire time it's in the funeral home/crematorium, according to my partner who works at one. It gets taped to the outside of the urn.

That's not foolproof though, the wrong ashes getting released to someone is rare but is usually easy to spot afterwards.

There's no way for you to 100% guarantee that the ashes are your loved one, except that there's no strong benefit to a crematorium doing that and heads would roll if they did.

There have been scandals where pets were cremated in bulk then the combined ashes divvied out, but not humans.

Here the tag is a ceramic tile that is put in the coffin before cremation. And i think it is verified by at least two people. But in the end, if enough people fuck up or fake, it could be anybody.

Snort a line and see what it tastes like.

Me tasting ashes: .... Definitely SatansMaggotyCumFart!
My friend looking at me: .... How do you know?
Me: .... it tastes funny.
Uuughhhh, this reminds me of that episode on My Strange Addiction with the widow who was addicted to eating her husband's ashes. Fucking heartbreaking, man.
That's not traditionally how humans go about tasting things, but I guess everyone's a little different huh? Do you eat through your nose too?

The important thing is to remember them, regardless of the contents of the urn, right?

Sure, but you can do that for free.

Paperwork and chain of custody. A reputable creamatorium is important.

There is a small chance dna would be present but it's a last resort.

When my grandpa was cremated, we went in the cremation room to see him for the last time. Then we stepped out. After a couple of minutes, we got back in to see the remains. Apparently there are still a lot of larger solid bones that remain, and they have to ground them up into the “ashes” you normally would see. We step out of the room again, and they come out with the urn containing the ashes after another few minutes.

Taste it. It's the only way to know for sure. Ofc, this is most helpful if one was forward thinking enough to have already taken a sliver of them during life, charred it, and then tasted that. It's kind of hard to compare without that data point, but, depending on how well one knew the subject, it might still be possible.

Taste test.