Adding the "a" makes it a somewhat stronger statement. It makes it sound like it is a little more important to you. As others have noted, "I'm Russian" is perfectly good English and neutral. I'm a Christian. I'm Christian. I'm a diabetic. I'm diabetic. I'm a socialist. I'm socialist.

When someone ask's you where you're from it's better to use the second one. Where are you from? I'm Russian. or I'm from Russia. Saying "I'm a Russian" would be very weird. It's hard to explain why, but basically it means "I'm a Russian person" which is unnecessary information. It would be common in this context: There's a Russian that works there. There are many Russians living there.

I am Russian / I'm Russian I am a Russian pianist.

I have brown hair And brown eyes

Thank you, Jazz.