Any hope of answering this question rests with an understanding of what emotions are. One way to look at it is from an evolutionary and biological point of view.
The modern human brain is the end product of numerous evolutionary steps, and that evolution is evident in its structure. Unique to humans are structures which are either smaller, far less developed, or absent entirely in other animals; these structures support logical reasoning. More ancient structures in the human brain, also present in all mammals, support behaviors associated with pair bonding and parental care. Even more ancient structures, present in reptiles, support behaviors including classic "fight or flight" responses.
The more ancient behaviors are associated with physiological/biochemical responses - the release of hormones such as adrenalin (fight or flight), and oxytocin (bonding), elevated heart rate, etc. No such response is associated with logical reasoning. These ancient behaviors/responses are what we commonly label as emotions. We perceive a distinction between emotional and non-emotional behaviors/responses because emotions are initiated by the more ancient structures of our brains, occurring outside (or beneath) conscious awareness and control. We are aware of them happening, but for the most part have no conscious control over their occurrence - we don't consciously decide to be angry or afraid. By the way, this confers a survival advantage because complex logical reasoning takes time and a "fight or flight" decision may need to be made and acted upon in a split second.
Computing devices as they exist in our modern world are entirely logical; Data is an extrapolation of our modern capabilities in computer engineering and robotics, so his behaviors (absent the "emotion chip") are predominantly logical. In creating Data, Dr. Soong conferred upon him a set of motivations - self-preservation, service toward humanity, self-improvement, and perhaps others, but Data's actions (again, absent the "emotion chip") always have a foundation in logic. By the way, there is a nod to Isaac Asimov and his "three laws of robotics" which imbues robots with a set of imperatives; Data at least implicitly follows these "laws". The "emotion chip" would seem to be about imbuing Data with some facsimile of what we experience as emotions - responses having no foundation in conscious or explicit logical reasoning. For what it's worth, "emotion" in Data could lead to behaviors in direct contradiction to any or all of "the three laws", or his normally "logical" motivations, and that can be seen in the "Generations" movie where his erratic "emotional" behavior interferes with his ability to fulfil his duty.
Absent the chip, Data had motivations, but depending on your chosen definition, didn't necessarily have emotions. His "desire to be human" isn't necessarily emotional... if he has a motivation for self-improvement, and logically reasons (never mind the path of that reasoning) that being "fully human" represents an improvement over what he currently is, then it could reasonably be stated (if not entirely accurately) that Data desires or aspires to be human, but without any emotional basis.
Also, let's not forget that Data's lines were written by fallible human writers who didn't always adhere to a show writer's guide and could always have slipped up when crafting dialogue. In at least one on-screen appearance, Roddenberry described Data as "Pinocchio" - an artifact which desperately desires to be human; his reference was somewhat flawed. Pinocchio was essentially a fully human mind/soul in an otherwise inanimate puppet - he craved a flesh-and-blood existence. Data sought to experience and understand "the world" as a human does, but from within his existing android body - to acquire what we call a soul.