This is an interesting situation you found yourself in because it should not happen the way you are describing.
I also at one stage tested both Ginger Wallet and Wasabi Wallet after reading a post stating they could be used to open the same wallet but not used simultaneously but I did not experience the issues you faced. When I ran Ginger Wallet it opened the correct wallet and when I closed it in order to run Wasabi Wallet, it opened the same/correct wallet too. The only rule to remember was they could not simultaneously.
I cannot be sure of the exact date but I tested this last year, maybe something has changed in the latest releases that caused the problem.
Furthermore,
I confess that I had several sync issues with my GingerWallet balance. Besides, my PC wouldn't allow me to install/run both software (Ginger and Wasabi) simultaneously, and there was some kind of conflict.
I had to uninstall Gingerwallet to install Wasabi. I really liked the idea of 2FA (Ginger) as an added security feature, but having my balance without sync issues made me switch back to Wasabi.
An important note is that since I already had a wallet created with GingerWallet, when I tried to open the same wallet using Wasabi, it was unable to open it, indicating the two wallet formats are incompatible between the two software. I had to re-enter my hotwallet's seed phrase in Wasabi.
As soon as I can, I'll test Ginger Wallet again and see if there are any improvements.