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The formation of institutions and the swelling of the congregation may take on a certain characteristic according to time and place and the casual observer may label it as "Hindu, Christian, Muslim etc"
The fact remains that we were born in this world alone developed a body and in time will pass away alone. We cannot take organizations or dress with us. The Idea that I am Hindu, Christian, Muslim, Man, Woman, American, Indian is a temporary designation only. The eternal soul (jiva atma) needs to wake up as to return back to the spiritual atmosphere (Vaikuntha) and gain association with the supreme lord and his devotees.
One does not need to be Hindu, Indian, European, or PHD. One simply needs to realize one's real self-interest and engage in devotional service under the direction of the bonified spiritual Master. This is called Sanatana Dharma (Eternal Natural Propensity).
Sri Chaitanya has taught us that our eternal designation is not this or that, but instead we should aspire to be the servants of the servants of Sri Krishna. Then we will truly be happy.