Here's the most obvious reason why: the current cultural construct of sexual identity (with it's accompanying categories of 'heterosexual' and 'homosexual') is a recent invention.
This is how Jenell Williams Paris explains it in her book The End of Sexual Identity: Why Sex is Too Important to Define Who We Are:
"Sexual identity is a Western, nineteenth-century formulation of what it means to be human. It's grounded in a belief that the direction of one's sexual desire is identity-constituting, earning each individual a label (gay, lesbian, straight, etc.) and social role. Perceived as innate and as stemming from inner desire, sexual identity has to be searched out, found, named and expressed in order for each person to be a fully functional and happy adult. Finding our sexual feelings is part of how we come to know ourselves and present ourselves to others.
Heterosexuality is a sexual identity category, and because people believe in it, it has real social and personal impact. It is also an idea that came from somewhere--it's a concept that has a history, albeit a relatively short one. Of all humans who have ever lived, very few have had sexual identities. Defined in a wide variety of ways, social identities related to sex (such as male and female) and gender (such as boy and girl) are common across world cultures. Identity categories based in sexuality (such as heterosexual and homosexual) are much less common. Most cultures that have ever been present on the earth, including biblical ones, didn't have heterosexuals. They didn't have homosexuals either, because heterosexuality requires homosexuality; each makes sense only with reference to the other. Like fraternal twins, they may not look alike, but they shared a common gestation." (p. 41)
A couple pages later, Paris further explains the biblical problem with heterosexuality:
"Heterosexuality is a concept riddled with problems. I'd even call it an abomination. The fact that heterosexuality is a social construction isn't what's problematic. We need identity constructs in order to function...It's even possible to 'Christianize' social constructs...The major problem for Christians with heterosexuality, and sexual identity in general, is that it is a social construct that provides a faulty pattern for understanding what it means to be human, linking desire to identity in a way that violates biblical themes. No pattern is perfect, but this one isn't even close. And 'Christianizing' sexual identity--whether by affirming or negating the morality of various sexual identities--doesn't help, because it doesn't address the faulty connections that sexual identity categories make between human desire and identity." (p. 43)
Here are a few helpful things I take away from Paris' points.
1. We have to be self-aware about the cultural constructs both of our own time and of past ages. This is why questions like "Is it a sin to be gay?" can be misleading. The very way the question is framed presupposes a cultural construct from our present time. This means that before answering the question, we need to do more work to unpack the assumptions behind the question. The Bible certainly speaks to sexual ethics today, but we have to be aware of how our own categories shape our reading of the Bible and how ancient categories shaped the biblical authors. Christians shouldn't be afraid to engage in this kind of cultural and historical analysis of the past and present.
2. Paris helpfully underscores that we have to analyze cultural constructs themselves. To say that something is a "cultural construct" is not a criticism. Humans are naturally culture-makers, and this includes cultural constructs with respect to gender and sexuality. So the question is not whether we are operating with cultural constructs, but how those cultural constructs are more or less in line with key biblical themes, as Paris points out.
3. Paris' observation cuts both ways in the current debate around same-sex relationships. 1st-century writers like Apostle Paul are not working with the concept of orientation. So some people think that nothing in the Bible speaks about (or against) loving, monogamous same-sex relationships. But those same interpreters often seem to assume that our cultural constructs are somehow timelessly true. Paris helpfully points out that it's one thing to talk about sexual attraction, but another thing altogether to connect our identity to that desire. Before we ask about what the Bible has to say to both gay and straight people, we need to question the underlying assumptions that give us the identity markers "gay" and "straight."
4. All this leads to the real reason I think Paul wouldn't claim to be a heterosexual, as understood in our culture. Our culture thinks sexual identity determines behavior: if I'm straight, that dictates what I should do sexually; if I'm gay, that dictates what I should do sexually. And of course, in a culture where sexual identity reigns supreme, the idea that anyone might be called to celibacy or singleness sounds ridiculous. But in Gal. 3:28, Paul relativizes the cultural constructs surrounding the terms "Jew and Gentile," "male and female," and "slave and free." All three of these terms refer not just to 'natural' differences but to cultural constructs that are identity markers and indicators of how one should behave. With respect to all of these identity markers, Paul says, "It is Christ, not the surrounding culture, who dictates how you are to operate within these roles." Jews eat with Gentiles because of Jesus; husbands love their wives as Christ loved the church; Philemon treats Onesimus as a brother in Christ, not a slave. In the same way, neither heterosexuality nor homosexuality determines how Christians ought to act in their sexual lives. I want to be clear: I don't think Paul would affirm same-sex eroticism of any kind. But he also wouldn't affirm probably about 90% (or more) of our culture's practices with respect to opposite-sex eroticism. All Christians are called to sexual lives of self-sacrifice, discipline, and holiness. All Christians are called to derive their identity from Christ and their ethics from Scripture, not surrounding cultural constructs. To the extent that some Christians have employed sexual identity constructs and used them to look down on or single out those who self-identify as gay, it's a problem. To the extent that some Christians have employed sexual identity constructs and used them to justify sexual activity that Scripture doesn't approve, it's a problem. All this simply means that we have more work to do in "taking every thought captive to Christ." (2 Cor. 10:5)