People often refer to Termina as some sort of dream or alternate dimension. here are my thoughts. There is nothing stating this outright.
The game's manual, the Hyrule Historia and developer comments all state, outright, that Termina is a parallel universe.
Skullkid can't possibly exist in both.
The Skullkid is the one who plays Saria's Song in
Ocarina of Time. He is from Hyrule and simply travels to Termina in the same way Link does. He can exist in both, of course, not at the exact same moment.
The game's characters look the same because the designs are excellent and the game was rushed through development.
I wouldn't say 'rushed' but yes, the reuse of character models was a time saving effort to ensure the game was finished in a short time-frame. However, I don't understand how that precludes Termina from being a parallel universe. It seems to support the notion by providing parallel versions of Hyrule's inhabitants.
You can clearly see Death Mountain from Termina.
No you can't. The mountain I suspsect you are referring to is Woodfall. Within the ring at the peak is Woodfall Temple and you speak to Kaepora Gaebora while stood on the side of the mountain at one point. It is not Death Mountain.
The official timeline places it as part of the timeline after Ocarina of Time and leading up to Twighlight Princess with no mention of dimensional tracvel. All of this seems to destroy the alternate dimension theory. What do you think
OoT makes mention of "the gap between dimensions", where Ganondorf banishes his Phantom after the Forest Temple is cleared and
Twilight Princess has the Twilight Realm which could be described as another dimension. The Zelda franchise has established other dimensions, even before
Majora's Mask, so I do not see how there has been no mention of them and how that destroys a fact the developers have confirmed several times.
1. Link is never seen leaving Termina; he continues his quest for Navi.
Just because we didn't see him leave doesn't mean he stayed in Termina. Hyrule Historia confirms that the Hero's Shade from
TP is the Hero of Time, the Link who was returned to childhood and went to Termina. That strongly suggests Link returned to Hyrule after
MM.
2. The game guide vaguely says it is a parallel world to Hyrule, that is likely a comparison that it is very similar to Hyrule. This statement could also have been easily mistranslated.
Parallel universe means another universe that runs alongside the universe Hyrule is in. Termina and Hyrule are obviously very different places, so I would strongly doubt the term 'parallel universe' is meant as a comparison. The fact that the same words were used again and again in regards to Termina would lead to the conclusion that it is not a mistranslation or a vague statement. The consistency with which it has been used tells us that it is an accurate and appropriate term to use with regard to what Termina is.
3. Anyway it seems odd dimension traveling would be never mentioned if it is so key to the story. It's never even suggested to actually occur.
The suggestion comes from the game manual when it says Termina is a parallel dimension. The suggestion is furthered at the start of the game when Link falls through a large hole with flashing, colourful images whizzing past and then walks through a mysterious,twisting corridor that warps around him. The game doesn't need to explicitly have some stand up and say "Link! You have travelled into a parallel universe!", we are simply introduced to the notion and then find things in-game that fits with that. Dimensional travel is not key to the story either, the story is about stopping the Moon and saving Termina, redeeming the Skullkid and solving people's problems.
Honestly, I see absolutely no reason to doubt that Termina is a parallel universe.