Suppose space is empty except for one single object.
According to relativity theory it doesn't matter whether we assign a constant velocity to it or not; in both scenarios the same is predicted.
The same cannot be said when comparing the object with a constant vs changing velocity - when assigned an acceleration, the object would feel something similar to gravitational pull.
Does it make sense to be able to assign an acceleration to the single object?
Answer
You are quite correct that for a single isolated object the velocity can be assigned any arbitrary value. That is because velocity can be measured relative to any other object so it doesn't have any single well defined value. However an isolated observer can measure their acceleration in their rest frame, and this is called their proper acceleration.
Strictly speaking the proper acceleration can still only be measured relative to some other object because the measurement requires to to drop something and see how fast it accelerates away from you. For example right now I'm experiencing an acceleration of 1G, and I know this because if I drop my pen it accelerates away from me (towards the ground) at 1G. The astronauts in the International Space Station are weightless because when they drop a pen it stays floating in the air next to them. However the proper acceleration measurement differs from a velocity measurement because it is relative to a well defined reference i.e. a freely falling object. That's why the proper acceleration has a well defined value while velocity does not.
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