tl;dr- This claim is true. It may sound weird that some biological males are castrated and then raised as though they were female, however it's been done to address a birth defect where genitals aren't properly formed. A recent survey suggests that this practice is in decline.
Other studies show that sex-specific biology affects behavior in ways that aren't reversed by genital reconstruction or social conditioning. These biological differences include hormone levels and brain structure.
Study
The central claim in this question is addressed based on
"Discordant Sexual Identity in Some Genetic Males with Cloacal Exstrophy Assigned to Female Sex at Birth" (2004) which isn't behind a paywall.
Other sources seem to agree, and these findings seem to be noncontroversial from a scientific perspective. Related studies can be found by searching for literature that cites this paper.
Related:
Why are males being castrated and raised as females?
The background section explains why biological males are being castrated. In short, it's to resolve a birth defect:
BACKGROUND Cloacal exstrophy is a rare, complex defect of the entire pelvis and its contents that occurs during embryogenesis and is associated with severe phallic inadequacy or phallic absence in genetic males. For about 25 years, neonatal assignment to female sex has been advocated for affected males to overcome the issue of phallic inadequacy, but data on outcome remain sparse.
–"Discordant Sexual Identity in Some Genetic Males with Cloacal Exstrophy Assigned to Female Sex at Birth" (2004-01-22)
Wikipedia describes this birth defect, cloacal exstrophy, as:
Cloacal exstrophy (EC) is a severe birth defect wherein much of the abdominal organs (the bladder and intestines) are exposed. It often causes the splitting of both male and female genitalia (specifically, the penis and clitoris respectively), and the anus is occasionally sealed.
So, what do doctors do when a patient has deformed genitals? As described in the paper:
THE CONCEPT OF SEXUAL IDENTITY IN persons with genital malformations has
intrigued the medical world since Money and colleagues' pioneering studies of intersex in the 1950s.1,2 They later reasoned that an infant's sex could be assigned if corresponding genitalia were constructed during infancy and the child's upbringing corresponded to that sex.3
–"Discordant Sexual Identity in Some Genetic Males with Cloacal Exstrophy Assigned to Female Sex at Birth" (2004-01-22)
The paper describes these patients as biological males with deformities rather than intersexed:
Cloacal exstrophy is not an intersex condition: aphallia and phallic inadequacy are structural anomalies.6-9,16
–"Discordant Sexual Identity in Some Genetic Males with Cloacal Exstrophy Assigned to Female Sex at Birth" (2004-01-22)
Study's results
However, it was unclear what actually happened to these newborns later in life; did the assigned gender stick after given the corresponding genitals and socialization?
RESULTS Eight of the 14 subjects assigned to female sex declared themselves male during the course of this study, whereas the 2 raised as males remained male. Subjects could be grouped according to their stated sexual identity. Five subjects were living as females; three were living with unclear sexual identity, although two of the three had declared themselves male; and eight were living as males, six of whom had reassigned themselves to male sex. All 16 subjects had moderate-to-marked interests and attitudes that were considered typical of males.
–"Discordant Sexual Identity in Some Genetic Males with Cloacal Exstrophy Assigned to Female Sex at Birth" (2004-01-22)
This finding seems to conclusively support Damore's claim that "Biological males that were castrated at birth and raised as females often still identify and act like males".
Gender assignment practices appear to be changing
It appears that the practice of raising biological males as female due to such structural issues is declining:
CONCLUSIONS: Although there is an association between the external appearance of the genitalia and the choice of sex assignment, there are clear temporal trends in this practice pointing toward an increased likelihood of affected infants being raised as boys. The impact of this change in practice on long-term health outcomes requires additional focus.
–"Changes Over Time in Sex Assignment for Disorders of Sex Development" (2014-09)
Biology is known to cause behavioral differences
It's probably common knowledge that behavior-affecting hormones vary significantly by sex. For example, adult males have about 20 times the the testosterone level of adult females. In popular culture, testosterone is often taken to characterize male behavior.
In women the testosterone levels which were only about 5% of that of men from the same age group decreased only slightly, starting from a median of 0.9 to 0.6 nmol/l.
–"Reference intervals for testosterone, androstenedione and SHBG levels in healthy females and males from birth until old age." (2005)
There're also differences in brain structure, and these differences do affect behavior:
During the intrauterine period the human brain develops in the male direction via direct action of a boy's testosterone, and in the female direction through the absence of this hormone in a girl. During this time, gender identity (the feeling of being a man or a woman), sexual orientation, and other behaviors are programmed. As sexual differentiation of the genitals takes places in the first 2 months of pregnancy, and sexual differentiation of the brain starts during the second half of pregnancy, these two processes may be influenced independently of each other, resulting in transsexuality. This also means that in the case of an ambiguous gender at birth, the degree of masculinization of the genitals may not reflect the same degree of masculinization of the brain. Differences in brain structures and brain functions have been found that are related to sexual orientation and gender.
–"Sexual differentiation of the brain and behavior" (2007-09)
Due to factors like these, it's unsurprising that biological males would still exhibit stereotypical male behaviors despite having had their genitals restructured and being raised as female.
Reference: What's sex?
In humans, sex is defined by the XY sex determination-system. Most other mammals use this same system, though it can be different for insects, fish, etc..
Almost all (~99.94%) humans fall into this system, being either XX (female) or XY (male). However, some people can have an extra chromosome in some of their cells, such as in Down syndrome. Since a person with XXX chromosomes still uses just the X chromosome while a person with XXY still has a Y affecting their chemical makeup, the definition of sex has been extended to:
People with a Y chromosome are male.
People without a Y chromosome are female.
- Almost all females are XX; but X, XXX, XXXX, XXXXX, etc., qualify.
People who can't really be said to be uniformly with-or-without a Y chromosome are intersexed.
The above study that this answer is based on didn't have any intersexed individuals; all males were male by the standard biological definition, not to be confused with gender identity.
DISCUSSION
Cloacal exstrophy is not an intersex condition: aphallia and phallic inadequacy are structural anomalies.6-9,16
–"Discordant Sexual Identity in Some Genetic Males with Cloacal Exstrophy Assigned to Female Sex at Birth" (2004-01-22)
So, that study demonstrates that unambiguously male infants, i.e. those with XY chromosomes, still retain male behaviors despite castration and being raised female.