Devotee of Vishnu is one who recognises Vishnu as the supporter of the entire existence. It is not the body that determines if one is a Vaishnava or not. Prahlada was born in the family of rakshasas, but is a great devotee of Vishnu. Not only a rakshasa, even a few animals such as Jatayu, Gajendra are regarded as great devotees of Vishnu.
The word Vishnu means one that spreads and makes everything part of its scope.
Similarly, in the universal body of God – Not everything exists for your purpose. Recognise this and respect the rest of the existence.
That is why, ‘Worship your own, Respect all’ is one of our principles / slogans. Ramanujacharya preached the Vedic literature in this right perspective after learning it and drawing conclusion in complete sense. Most works of other scholars do not include this level of comprehensiveness.
Therefore, everyone is a Vaishnava. Some recognise this fact, and many others do not.
-From the discourse of Sri ChinnaJeeyarSwamiji