Wolverine's Dog Tag was a military ID tag that belonged to Wolverine. It is a necklace showing a number which is 45825243-T78-A.
The tag was given to James Howlett just before his skeleton was bonded with adamantium at Alkali Lake, and he chose the epithet "Wolverine" based on Kayla Silverfox's story. He kept it following his memory being erased by Stryker, and throughout his aimless wandering in the Canadian Rockies it served as the only reminder of his past life.
Years later, after joining the X-Men and thwarting Magneto on Liberty Island, Wolverine gave it to Rogue for safekeeping, saying he'd be back for it one day, while he searched for clues to his past and real identity.
Upon escaping the X-Mansion following William Stryker's military assault on the premises, Rogue gave his dog tags back to him, while they fled to Boston to meet with Storm and Jean. After regaining part of his memory about the Weapon X program at the end of X2: X-Men United, Wolverine threw the tags on the ground next to William Stryker, who was chained to a brick wall near a collapsing dam. The dog tag, along with Stryker, were ultimately washed away after the dam collapsed.
In the revised timeline, it's shown in Logan, that Logan still has his dog tags, now appearing darker and rustic showing their age. He leaves them on his nightstand in his room, and never retrieves them when he flees from the Reavers.