As I tried to say before, our kihon waza of tanto dori / jo dori / tachi dori irimi nage ends with tori immobilizing uke by applying a choke. It is called "choke" because it is about interrupting uke's possibility to breathe. It is endet when uke taps out.
I wouldn't call this a pin? But it's clearly a form of katame waza just like the finalization of shiho nage or kote gaeshi can be.
Also there is a form of tachi waza irimi nage where uke is not thrown but choked, tori standing behind him. It can be applied as headlock, but also as choke. We do this very seldom. Here also: It is endet when uke taps out.
In both cases I don't know whether to call it 'formal'? It simply "exists".
Interesting. I suppose it depends on your purpose really. All the above rely on airway blocking or even blood supply blocking thus I can say I don't teach or put that into practice as I consider it unnecessary.
I have on the other hand taught similar without the need to restrict any blood or air flow.
Thus I leave it to judo or other forms of Aikibudo or Aikido to practice in that way.