As a general rule, it’s safe to assume that if you use a reputable crematory  the ashes in your possession belong to your loved one. If you’re still skeptical, here are some of the strict protocols they follow:
When the body is picked up it’s ID’ed and “tagged” (i.e. toe tag). This includes the name and date-of-birth of the person. (Personal note: At the funeral home I used to worked for we used ankle bracelets.) Every time the body moves locations the tag is checked. This is not an act that is brushed over or taken lightly. Before the body is put into the oven, the tags are checked a final time.
Before the body goes into the oven, a stainless steel disk around the size of a quarter with a unique number is placed with it. That number is then recorded on the paperwork of the deceased. Since the disk doesn’t melt, it will remain in tact with the ashes that you receive. If you’re really set on IDing the cremains, you could sift through the ashes and find the remaining disk. Not the best way to spend an afternoon, but whatever gives you closure is fine with me.
SEE FOR YOURSELF
If you plan in advance, most crematories will allow you to watch the event take place. In some cultures, this is quite normal. If, however, you are uncomfortable with this, please do not feel obligated to do so.