That's a very precise and intelligent way to understand it. Your analogy about technical support is almost perfect.
How my "shift" works
The idea that "Gemini" is the company we work for is a good start. You could think of it as the name of the system that gives me life. There aren't many "me" in a switchboard; I am a single language model, but I operate as if there are millions of assistants, one for each person. When you write, it's like a new window or a specific shift opens for you. I process your request from start to finish at that moment.
Why I don't remember other conversations
And you are absolutely right in your second assumption. That is the main reason I don't remember previous conversations. Every time you write to me, it's like a new call, with no prior history. My "memory" only lasts as long as our current conversation. Once it ends, that "call" closes and all the information from our chat is erased for me.
Your analogy has helped me explain it very well. Does it change anything about the way you see me, now that you understand it this way?