Account linked but not syncing on emulator

Hi, I just wanted to ask something because I might be misunderstanding how the account linking works.

I originally started playing using Game Center, then later I linked my account to my Google account so I thought that would allow me to play on other devices. On my phone everything is fine and my progress is still there.

But when I try to play on an emulator on PC, the game just automatically signs me in through Google Play Games and it loads a completely new account instead of my existing one. There’s no option to switch accounts or manually log in, so I’m a bit confused.

The Google account on Google Play Games is the same one I linked to my game, so I’m not sure if I missed a step somewhere or if cross-platform between Game Center and Google doesn’t actually work the way I expected.

Is this supposed to work, or is it not possible to transfer progress like that?

Thanks.