When I first saw this episode of TNG, I thought that such a language was impossible. Without even thinking about how often I hear and use phrases that likely make zero sense to someone who doesn't know the same context as myself. Like dealing with someone who lies all too often, "they are the boy who cried wolf" gets used. Meaning, one of these days they are going to tell the truth but no one will believe them. Or saying, "we need to touch base" to someone who isn't familiar with baseball. Sometimes I understand the meaning of a phrase, like "the proof is in the pudding," without knowing what that phrase originally meant. I still wonder what pudding? every time I hear it. So, yes. I can believe the Tamarian language. Because context is important.
I don't think the Tamarian language as a whole is represented in the episode. I can't imagine "Temba, his arms wide"-ing a way to warp capabilities. I think of the Tamarian language shown as a universal shorthand for the Tamarians. Like their society has different languages, but the Tamarian we hear is the common language. That Tamarians specializing in a field of study have their own words and phrases, much like on Earth of today. For example, if I listen to a lecture on biochemistry, I wouldn't understand most of it because I don't speak (understand) biochemistry. I have no context.
I think the Tamarian language is fascinating, and possible.