Sleep is meant to be applied intuitively
Sleep is not defined as a condition or state in the PHB or DMG (the Dream spell was written before the optional rules on sleep in Xanathar's Guide to Everything). Rather, from the DMG "Using and Tracking Conditions" (p.248 - emphases mine)
Various rules and features in the game are clear about when they apply a condition to a creature. You can also apply conditions on the fly. They're meant to be intuitive for you to do so. For example, if a character is in a state such as sleep, that lacks consciousness, you can say the character is unconscious.
This passage does not say that sleeping characters have the unconscious condition. Rather, the point is that sometimes a DM can apply the mechanical effects of conditions 'on the fly', that is, when they feel the situation as such warrants it. In the case of sleep, it is not that a character is unconscious, but rather that they lack consciousness and thus can be treated for some situations as if they were unconscious, while for other situations they may be treated as if they are not unconscious (for more on this, see my answer to What is the difference between "Unconscious" and "Asleep"?).
Nothing in the Dream spell implies that it forces a creature to remain asleep, but:
On a failed save, echoes of the phantasmal monstrosity spawn a nightmare that lasts the duration of the target's sleep
So the nightmares will last the duration of the sleep, but nothing in the spell says that the target must remain asleep for the duration of the spell. Since sleep is supposed to be applied intuitively, it comes down to your DM's sense of what seems naturalistic. In our world, sometimes people wake up from their nightmares immediately, starting awake. Sometimes people have the sense of having had nightmares 'all night long'. Sometimes people have 'night terrors' where they are conscious of being asleep and desperately trying to awake but cannot. Any one of those could apply at the DM's discretion, and it would equally be the DM's discretion as to whether the target could make a conscious effort to awake.
I suggest that PC's be given choices
In this DM's experience, players enjoy the game more when they are given more choices about what happens to their character, while being given fewer choices often leads to resentment. In the case of Dream, there is an important trade-off:
On a failed save, echoes of the phantasmal monstrosity spawn a nightmare that lasts the duration of the target's sleep and prevents the target from gaining any benefit from that rest. In addition, when the target wakes up, it takes 3d6 psychic damage.
Staying asleep longer staves off the damage but means more of the rest is wasted. Waking sooner means taking the damage sooner but then having more time in which to re-start their rest. If this spell targeted a PC in my campaign, my inclination would be to give the player the choice of when to wake up; 'you can force yourself awake, taking damage now but then having a chance to actually have a long rest before the morrow, or you can toss and turn with bad dreams all night, not getting a rest but at least not taking damage until your healers have had a chance to rest themselves and replenish their spells'.