I think it depends your definition of 'real'
Elvish is a language with a fully constructed vocabulary, rules of grammar, a numerical system, a written form, and even dialects (Quenya, or 'high elven', and Sindarin, the more common form)
Does being constructed in one lifetime instead of over millennia make it not real?
Does not being an official language of any country make it not real?
I think real is an appropriate word to use; it's just an unofficial, more recently constructed language.
When LOTR was in theatres, my older sister took me to see every one, and we had elvish lessons. We can still carry on simple conversations in it
Vendui! (Hello!)
Vendui, mellonamin, revanima, n'uma ta? (Hello, my friend. Beautiful day, isn't it?)
Uma! Lle'na quel sinare? (Yes it is! Are you well today?)
Uma ta'amin, diola lle (I am , thank you)
I consider it no different than Spanish or old Latin, just more exclusive