yoga as a system of philosophy (Patanjala Yoga Sutra is the leading standard treatise) got integrated into Hinduism..
and that way can be called part of that religion..
But Hinduism with its theistic and ritualistic hues is what we find as a religion these days,
and Mimamsa of Jaimini and Advaita of Badarayana,,, and the later Bhakthi cults seem to guide that religion at present.
Yoga in sum and substance is a way of life.. guiding us to channelise our thoughts and physical action( this physical element is less pronounced traditionally,
but the West generally understands only the physical postures and exercises as yoga,,)
and here there is hardly any Hinduism or for that matter any religion involved
Yoga is part of Hinduism only to the extent that it does not negate the authenticity of Vedas..
You can find Advaitins, Dwaithins, Visistaadwaitins, followers of Bhakthi cult, Tantriks, Shakthas Shaivas etc in Hinduism,, but there is hardly anyone who uses the yogasutras as a spiritual text..
Of course all of them and non Hindus too can easily follow the concepts of yoga
Yoga is completely secular.. it has no caste creed or religion attached to it in reality..
But if some particular group assumes the parenthood, others need not accept such parentage.