Okay, look, people's mental health and psychological suffering probably is not the best topic for stochastic arguments in a "debate club" style, where you can nonchalantly pick a pro or a contra statement and than come up with random arguments that play with definitions and semantics. Your question is based on semantic nitpicking, but transgenderness is rather a material and biological phenomenon with a certain mechanism.

Scientific researches have shown that there are 3 brain areas that in transgender people even before HRT have the same configuration (size and innervation type) as in cisgender people of their psychological gender: the bed nucleus of stria terminalis in the extended amygdala (Zhou et al., 1995; Kruijver et al., 2000), the third interstitial nucleus of the anterior hypothalamus (Garcia-Falgueras et Swaab, 2008), and the putamen of the lentiform nucleus (Luders et al., 2009). Those brain areas are gender-specific, they have different structure in cis men and cis women, and since they belong to the limbic area that is known to be responsible for emotions and instinctive behavior, a lot of researchers believe that they are responsible for sexual instinct and gender-specific behavior.

A simple formulation is that transgender people have gender-sexual instinct of their psychological gender. That makes them instinctively want to participate in sexual competition among individuals of their psychological gender, and makes them feel psychological pain and frustration when they cannot participate and win it, cause they don't have sexual characteristics of that gender and cannot act or be treated by their conspecifics according to their sexual instinct.

Also, there have been experiments on "transgender" animals (rats, guinea pigs, rhesus monkeys) who demonstrated sexual behavior of the gender opposite to their genitalia gender (after being treated with the sexual hormones of the opposite sex at the prenatal or neonatal stages). So, physically-female rats developed male sexual behavior: mounting and pelvic thrusts; and physically-male rats — female sexual behavior: female lordosis when being mounted by a male rat. Researches also showed that treating those rats with the hormones corresponding to their genitalia gender later in adult life didn't cancel the effects of the prenatal hormonal treatment.

Hines, 2006:
In both rodents and nonhuman primates, exposing developing female animals to high levels of testosterone, for example, by injecting them or their pregnant mothers with the hormone, increases subsequent levels of male-typical behaviour (1). In rats, for instance, a single injection of testosterone on the day of birth produces female animals who show elevated levels of rough-and-tumble play, a behaviour that is normally more common in juvenile males than in juvenile females. Similarly, female rats treated with testosterone neonatally show increased capacity for male-typical sexual behaviour and reduced capacity for female-typical sexual behaviour in adulthood. Removing testosterone from developing male rats (e.g. by neonatal castration) has the opposite effects, reducing male-typical behaviour and increasing female-typical behaviour. Comparable effects, on both juvenile play behaviour and on sexual behaviour in adulthood are seen following prenatal testosterone manipulations in rhesus monkeys (3, 4).

Phoenix and his colleagues (1959) demonstrated that administering testosterone propionate to pregnant guinea pigs increased the chances that their female offspring displayed male sexual behavior as adults, but exacting the reverse for female sexual behavior. Although Phoenix and colleagues suggested that prenatal androgen treatment permanently changes neural circuits involved in sexual behavior, they did not expect that such changes would be as dramatic as those observed in reproductive organs.

Those observations lead us to a conclusion that transgenderness probably is based on animal instincts and hormonal balance at the prenatal stage of individuals' ontogenesis.

So, going back to your question, it seems like such feelings (feeling like a person of a certain gender) are not based on culture and acquired behavior, so they don't depend on being raised in a certain way.