Does it mean that the dog tag in the game (Summoner's Canyon) can't be displayed with a/t+5 or the summoner's avatar icon can't be displayed?

If it is a game, it is because the version and time are wrong. In that version, only teams participating in the season are allowed to use dog tag special effects. For example, if you take part in S6 competition now, you can display the dog tag by buying SKT icon avatar. After the version and the game, only team avatars can be obtained, and dog tags cannot be displayed.

If the summoner's avatar icon can't be displayed, it may be that you accidentally deleted this avatar icon when cleaning up the garbage. You can use the game to repair or download the game again ~