I just bought an older 16 gauge 870. The previous owner had a new full choke barrel installed with an internal screw in choke. Don't know if It was factory installed or done by a gun smith. It came with one screw in modified choke. I want to buy an improved cylinder choke. My questions are number one, how can I tell if this a Remchoke, Winchoke, or some other so I can match the other choke? Question number two since the fixed choke is a full choke can I take the modified choke out and use it as a full without damaging the barrell? I have never owned or used a gun with interchangeable chokes. Help please.

Being as it is 16 ga finding another one to compare it with might be tough. You can mail it to Briley (www.briley.com ) and they can tell you what it is and sell you another choke tube.
It is not recommended to shoot a choke tube threaded barrel without a choke tube installed. Yes, I know people do it. If you shoot without a choke tube installed you will have NO choke; it will be cylinder bore. Since your barrel has a choke marking on it (Full) your tube installation is almost assuredly aftermarket. When they install tubes they have to bore out all the old choke plus a little first.

Go to Carlson's website or TruLock Chokes (https://trulockchokes.com/) and use their identification tool to match yours up. You may have to send it to them.