When all was said and done, Jon Snow's story on Game of Thrones ended up being surprisingly cyclical. He started the show by joining the Night's Watch and ended it by doing the same. The first time it was voluntary, but in the finale it was his punishment for killing Daenerys to save the kingdom from her tyranny. But he didn't stay put at Castle Black. It seems like Jon became a wildling on Game of Thrones, or at least decided to journey farther north with them.
Jon did not look thrilled to be sent back to the Night's Watch, where he would be watching a Wall that was already (mostly) toppled by the Night King and keeping an eye out for a threat that no longer existed. The fortress was filled with wildlings, which was a little different from the Castle Black of years past. After strapping on his sword and giving Ghost a long-awaited pat, Jon hopped on a horse and headed beyond the Wall. Tormund, Ghost, and the rest of the wildlings accompanied him, but no other Crows could be seen. There wasn't any dialogue, either. Jon didn't announce his intentions to join the Free Folk or put on their telltale furs. That makes it impossible to know exactly what Jon decided to do.
Before he left for Castle Black, it seemed like Jon would be honoring his renewed vows as a Night's Watchman. He told Arya that she could come visit him there, so he must have been planning to stay put, at least initially. But when she revealed that she wouldn't be returning to the North again, perhaps Jon decided that he had no reason to linger. During his ride with the wildlings, he did pause to look back at the gate significantly, and it's possible that was supposed to be a silent goodbye to everything he was leaving behind.
Jon did express a desire to go North with Tormund before the siege of King's Landing. During their romantic farewell in Episode 4, Tormund said that he thought Jon would be happier beyond the Wall because, in his words, "You've got the North in you. The real North." Jon was certainly tempted, and it seems like he decided to follow his bliss in the finale. He wanted to return to the life he knew in the far North. Whether or not he became a wildling to do so is inconsequential. He was ready to leave Westeros behind, so he did.
After all the talk of destiny and parentage, it felt anticlimactic for Jon Snow to end the series like he did. He was conflicted and sad for several episodes, killed his aunt-slash-girlfriend as politely as he could, and then languished in a cell offscreen until he was banished. There was very little of Jon's perspective on the emotional fallout of his actions. But, ultimately, he did get what he wanted. Jon wished he was going with the wildlings in Episode 4, but it just took him a little bit longer to make that dream a reality. I hope he and Tormund are very happy together.