The following is an inference based on the information you provided:
There is a number "200" in the model:
This indicates that the graphics card may belong to AMD Radium Dragon HD 2000 series.
This is the card purchased on 20 15, except after 15:
AMD Radium Dragon HD 2000 series was launched around 2007, which meets the time requirement before 20 15.
No external power supply, no cooling fan, black pure metal cooling, decorated with red claw marks:
These functions may be the unique design of a specific brand or model, and not all AMD Radium HD 2000 series graphics cards have these functions. However, the design without external power supply and cooling fan usually appears on low-power graphics cards, which is consistent with some low-power models of HD 2000 series.
Poor performance. At 800x600, the Assassin's Creed Black Flag has only a dozen frames, and the nuclear display level is:
The performance of AMD Radium HD 2000 series graphics cards is relatively low, especially when dealing with newer games. Running "Assassin's Creed: Black Flag" at 800x600 resolution has only a dozen frames, which is consistent with the performance level of this series of graphics cards. To sum up, although it is impossible to determine the specific model (because different brands and models of graphics cards may have different designs), according to the information you provided, this graphics card is probably a low-power model in AMD Radium HD 2000 series. In order to find a graphics card with similar memory to yours, you can try to search the pictures and materials of AMD Laser Dragon HD 2400 or Laser Dragon HD 2600 to see if there are any styles that match your memory.