If Yoga were like fine wine, Hatha Yoga would surely be the wine for export. It could be said that Raja Yoga and Kundalini Yoga got a fairly strong foothold outside India, as well. All of these forms of Yoga have many aspects, but the physical aspect is what lures most Yoga students - outside of India.
Bhakti Yoga, on the other hand, is a spiritual form of Yoga and is very commonly practiced within India. Bhakti Yoga is union by love and devotion to God. Therefore, this is the form of Yoga that really does have a deep spiritual aspect and a connection to Hinduism. This spiritual, and religious, aspect is possibly the reason why Bhakti Yoga has not managed to proliferate as well as Hatha Yoga, which is Union by physical mastery.
Could Bhakti Yoga be practiced by a Yoga practitioner who is not a Hindu? Let's ask, for example, could a Christian, Moslem, or Jew practice Bhakti Yoga and stay within the confines of his or her religion? Some already do.
This could be considered sectarian Yoga, when a number of Yoga practitioners, of the same religion, gather to worship, meditate, pray, sing, or chant. Attachment to God is not reserved to any one specific religion. Therefore, in principle, Bhakti Yoga can be practiced by members of any religion.
There are many more types of Bhakti classifications, but I will name two basic classifications. Nishkamya Bhakti is praise to God for all that you have. This is appreciation and acceptance of all your situations, relationships, and everything else. Instead of being a seeker, you are now a "finder" of all that is good in your life. In turn, you will easily be able to project loving kindness to others.
In contrast, there is also Sakamya Bhakti. This is devotion to God with the desire for gaining something. The objective could be material gain, to free yourself from an ailment, or to gain power of some kind. In time of need is when most people pray to God. This is a common thread in all of us, and surprisingly our prayers are answered in a positive way.
It is important to understand that Bhakti Yoga is much more complex than this simplified explanation that I have presented. However, the principles of Bhakti Yoga are there for all Yoga students of every religion.
Interestingly, I have heard strong arguments against "oneness" with God. The theory is that God will only let us so close, as we are imperfect. It is true that humans are imperfect, but there is no harm in self-improvement or Yoga.