how can I activate my Apple TV 2nd generation (2010)?

The older models no longer work with iCloud. Even then, they only used server based channels and most providers have dropped support. So while some may still have working units there isn't much you can do with it and due to age it will be more common for hardware issues to arise. Even a used 4th gen will be a better bet as it will allow most of what you can do on a current model.

The 1st gen already ran a modified version of MacOS and had a sizeable HDD that synced media from a computer and had accessible storage.

The 2nd gen was completely redesigned and used a basic version of IOS and was streaming only, using a small flash drive for caching. It uses the same chip that shipped with iPhone 4 and the hardware was designed specifically for streaming server based channels with basic templates.

The 3rd gen continued with a similar OS (version 7) but the 2nd gen hardware was not capable of updating past version 6.

A completely new platform was developed for the 4th gen, TVOS, which was more in line with IOS and could run native apps. A proper storage system and space was added to accommodate this. It would not be possible to install on older hardware.

As such, there isn’t much you can do with 2nd gen now.