Philip Pullman's His Dark Materials explores several parallel worlds, including some where people's souls manifest as corporeal, animalian companions called dæmons. Most of the characters who live in these worlds have dæmons of a different gender than themselves, but there are brief mentions of people with same-gender dæmons in Pullman's books. Fans have theorized these characters are part of the LGBTQ community, which Pullman has neither confirmed nor denied.
In Lyra's world, everyone has a dæmon who can change forms throughout a person's childhood, but when puberty hits, their dæmon "settles" into a form that best represents who they are. People and their dæmons are rarely apart, because it causes them incredible pain to separate. However, it's taboo to touch another person's dæmon unless two people are in an intimate relationship. Even then, it provokes a startling sensation in person and dæmon alike.
There's nothing sexual between a human and their dæmon, though touching another person's dæmon clearly has a sexual component. Furthermore, dæmons settling indicates the transition from childhood to adulthood, when people's bodies begin to change and they potentially start exploring sex and intimacy. Characters having differently-gendered dæmons in His Dark Materials seems to be the norm, and the correlation indicates that a dæmon's gender could coincide with a person's sexuality and/or gender.
For decades, readers have questioned how queer or trans people's dæmons might manifest or settle. Pullman said in 2000, "Occasionally, no doubt, people do have a demon of the same sex; that might indicate homosexuality, or it might indicate some other sort of gift or quality, such as second sight. I do not know. But I don’t have to know everything about what I write."
In the 20 years since, Pullman has had even more questions hurled at him about Lyra's world, particularly in regards to LGBTQ representation. In 2018, he drew equal parts concern, ire and compassion from fans when he tweeted about "finding the trans argument impossible to follow." However, it appears his question was in good faith, because he has begun responding to fan theories and inquiries about LGBTQ characters with due consideration and care.
Even with the popularity of HBO and BBC's serialized His Dark Materials adaptation, Pullman has not confirmed whether queer or trans characters have same-gender dæmons, but fan theories abound. Some believe trans characters' dæmons match the sex they are assigned at birth, particularly if they are trans and straight. Some believe non-binary characters' dæmons take on forms with fluid presentations. Similarly, fans have theorized that gay characters who don't come out until later in life may have same-gender dæmons, potentially marking an outward indication of their sexuality from birth.
Canonically, there is no firm answer to whether or not LGBTQ people in Pullman's world have same-gender dæmons. Even if he establishes rules for this part of the His Dark Materials world outside of the actual books, those rules technically won't be canon. They would have to be made clear in the text for that to happen. In the interim, readers and viewers can come to their own conclusions about how they might experience this world, wherever they fall on the gender and sexuality spectrum.