There are some people who think that homosexuality is a choice. They would argue that we can make this choice because we have free will. I will focus on those who think this way because of religious reasons. Those who think that God gave us the freedom of choosing right or wrong. But God is also the creator of the animals. So why do most of the animals manifest homosexual behaviour? They do not have free will. They cannot chose, they do not know what right or wrong is, they just follow their insticts. Why would a God who thinks homosexuality is a sin, create animals who have homosexual relationships?
In the case of the Hebrew Bible/Old Testament, there is no inconsistency, because sin is restricted to humans. At the bottom of this page is a list of all HB occurrences for 'sin' (חטאת). Also interesting would be Gen 6:5 in the story of Noah:
The LORD saw how great the wickedness of the human race had become on the earth, and that every inclination of the thoughts of the human heart was only evil all the time. (NIV)
Even though god regrets having made "the human race ... and with them the animals, the birds and the creatures that move along the ground", wickedness (רע) is only reserved for humans (v. 5).
So, abstracting, you see that there does not need to be an inconsistency if you reserve the concept of sin for humans.
Whether the Jewish/Christian god thinks homosexuality is a sin is another issue that is not on topic here.
This answer is from a Judeo-Christian perspective, as that seems to be the basic perspective you're asking from. I don't know enough about the religions with reincarnation, and your assumptions about animals' lack of free will or rationality may not be true for them.
Firstly, sin is generally considered to only be a category that applies to sapient beings: humans and angels/demons. But while animal homosexuality may not be sinful, God could consider it to be unnatural, or counter to his designs.
A core concept of Judeo-Christian thinking is that the world we know has been holistically corrupted by sin. Humanity was created as a priviledged species, the "image of God", which both means that humanity is the most God-like of all created things, as well as indicating a role of being God's representatives on this earth. Whether sin has corrupted all things directly through the original human acts of sin because of those humans authority overt the world, or whether God more actively "cursed" the world, the result is that the world we know now is not the original perfect world of God's designs.
An example of plant thorns is given in the scriptures: God is said to have changed plants after the fall of Man by making some of them produce thorns, as a consequence and punishment for sin. So they are both natural from our perspective and unnatural from the perspective of God's original design. But God is still seen to be in control, and it is accurate to say that God willfully makes plants with thorns.
So within a Judeo-Christian perspective there is an inherent contradiction between what we observe as "natural" and what God would consider "natural". After sin "natural" has been changed. If God did consider animal homosexuality to be unnatural, it could be seen as one of the changes introduced by sin.
In the same way, this makes the whole argument over whether human homosexuality is a matter of choice irrelevant. If human homosexuality has genetic causes, that doesn't tell us anything about whether it is part of God's original design, because it too could be an unnatural change like plant thorns. In my opinion then, debates (within Christianity) over morality of human sexuality shouldn't really be about what is "natural", for it doesn't really tell us anything. But demanding that it is a matter of choice to people who don't feel like they do have a choice is sure to very insensitive and hurtful to many.
Using the focus you have given, even without questions of right or wrong the underlying choice is to do or not do. All people have this choice in every action in every moment. Staying with the focus, it is not thought that animals have the intelligence to choose to do or not do, free will, as humans do and that they act with animal instinct rather than choice.
Interestingly, the commandment requiring humankind not to sin was given to humankind, not animal kind. It can be argued that the animals do not sin since they do not have either knowledge or commandment upon them. But, anyone practising 'strange animal-husbandry' would be guilty of sin.