While many have tried to tie the 2 concepts together, Gender Identity is an entirely different concept to Sexual Orientation.
While you are more interested in scales for gender identity, as you are asking about both, I will give information on both.
Sexual Orientation Psychometric Scales
There are a few psychometric scales for sexual orientation. The most famous, I would say, is the Kinsey Scale. It was first published in Sexual Behavior in the Human Male (Kinsey et al. 1948), and was also prominent in the complementary work Sexual Behavior in the Human Female (Kinsey et al. 1953).
The Kinsey scale or Kinsey's Heterosexual-Homosexual Rating Scale measures sexual orientation, from 0 (exclusively heterosexual) to 6 (exclusively homosexual).
||Predominantly heterosexual, only incidentally homosexual
||Predominantly heterosexual, but more than incidentally homosexual
||Equally heterosexual and homosexual
||Predominantly homosexual, but more than incidentally heterosexual
||Predominantly homosexual, only incidentally heterosexual
There is the Shively & DeCecco Scale (Shively, & DeCecco, 1977), which examines, what they believed, are the four components of sexual identity:
- biological sex,
- gender identity,
- "social sex-role" — see also, the Bem Sex Role Inventory (Bem, 1974), and
- sexual orientation.
There is also the Klein Scale — also known as The Klein Sexual Orientation Grid (or KSOG). Introduced by Dr. Fritz Klein in the first edition of his book The Better Option (Klein, 1978), the KSOG is a system for describing a person's sexual orientation in a way that is more detailed and informative than previous methods.
Note that the Klein Grid is not a diagnostic tool intended to test and reveal a person's sexual orientation. Instead, it is a learning tool to help people better understand the nuance and complexity that make up everyone's sexuality.
Then there is Randall Sell's Scale (Sell, 1996).
Sell points out that measures of sexual orientation are categorized as:
- bipolar (such as the Kinsey Scale),
- multidimensional (such as the Klein Scale), and/or
- orthogonal (such as the scale proposed by Shively and DeCecco).
[T]he Sell Assessment of Sexual Orientation, is proposed based upon this review. Finally, methods of scoring and summarizing the proposed measure are discussed.
Gender Identity Psychometric Scales
Although there was Bem's 1974 characterisation of people as masculine, feminine, or "androgynous" through her BSRI - Bem Sex Role Inventory (Bem 1974); Harry Benjamin's Gender Identity Scale (Benjamin, 1966) was the first attempt to create a psychometric scale for gender identities.
Published in his book The transsexual phenomenon, it was extremely biased for many reasons, covering personal biases of the researcher, biases within society and therefore biases within the law.
As Psychology Wiki points out
Dr. Benjamin's Scale references and uses Dr. Alfred Kinsey's sexual orientation scale to distinguish between "true transsexualism" and "transvestism". But it should be noted that the strict relationship between gender identity (Benjamin's Scale) and sexual orientation (Kinsey's Scale) was just a result of the researcher's biases, not his scientific findings.
At the time when Dr. Benjamin practised, no transsexual could qualify as eligible for transition if he/she wasn't completely homosexual (relative to his/her birth sex), thus, heterosexual after transition. It was just because at the time it was believed that obviously a real, normal female should be completely heterosexual, and a lesbian female was perceived as somewhat deviant, wrong or abnormal.
Bem, S. L. (1974). Bem Sex Role Inventory [Database Record]. APA PsycTests. https://doi.apa.org/doi/10.1037/t00748-000
Benjamin, H. (1966). The transsexual phenomenon. Ace Publishing Company.
Kinsey, A. C., Pomeroy, W. B., Martin, C. E. (1953) Sexual Behavior in the Human Male, Philadelphia, PA: W.B. Saunders
Kinsey, A. C., Pomeroy, W. B., Martin, C. E., & Gebhard, P. H. (1998). Sexual behavior in the human female. Indiana University Press.
Sell, R. L. (1996). The Sell assessment of sexual orientation: Background and scoring. Journal of Gay, Lesbian, and Bisexual Identity, 1(4), 295-310. https://doi.org/10.1007/BF03372244
Shively, M. G., & DeCecco, J. P. (1977). Components of sexual identity. Journal of homosexuality, 3(1), 41-48. https://doi.org/10.1300/J082v03n01_04