EDIT
This question has been solved for the OP in the comments by a better way.
Anecdotally, there are sometimes differences between the (recommended) approach of Linux_Steam>Wine>Game and the (non-recommended) approach of Wine>Windows_Steam>Game. I suspect these differences -could- be overcome by working harder to configure things correctly inside Linux_Steam, and that this is largely about the convenience of which default settings end up being automatically applied.
I personally can't get my PlayOnLinux to install SteamSetup.exe on a virtual drive at the moment, but the following solution might be helpful to others in the OP's position, given it was already adopting a non-recommended approach.
Given how complex Steam is, Windows_Steam works incredibly well in Wine, so the OP might want to try creating a 32-bit Wine instance and installing SteamSetup.exe there - which takes out PlayOnLinux. I think that reducing the complexity is still virtuous when we are doing something non-recommended.
On my PC this would look like downloading SteamSetup.exe to some folder, then opening a terminal there and running the command:-
$ WINEARCH=win32 WINEPREFIX=~/.wine32 wine SteamSetup.exe
(But don't paste this into Terminal before knowing the right location of your wine Prefix and making sure that 32-bit not 64-bit is desired)
In that command, ~/.wine32 is where in the past I have asked wine to put my 32-bit wine prefix. On my PC, Windows_Steam is installed at ~/.wine32/drive_c/Program Files/Steam/ (btw I suppose that ~/.wine32 is a "prefix" and ~/.wine32/drive_c/ is a "virtual drive", and so a wine prefix is analogous to a Windows PC's 'My Computer' folder.)
When following this approach, 32-bit and 64-bit versions of wine have to be installed to separate wine prefixes, which from Steam's perspective are then counted as separate PCs for their user authentication.
Also, I've always found this to be problem-free but it might be wise to give Steam its own wine prefixes - in case it breaks them. I've trusted that the Prefix should be safe from errors inside Steam; Wine should be safe from anything that goes wrong inside a Prefix; and the Linux system should be safe from anything that goes wrong inside Wine - but if some bug or exploit in Windows Steam arises, those presumptions are all at our own risk. Of course often Steam accounts are valuable and linked to Paypal accounts.
For creating the wine prefix in the first place, there are lots of places to find instructions, e.g. https://www.addictivetips.com/ubuntu-linux-tips/create-wine-prefixes-on-linux/
In practice, PlayOnLinux is sometimes only automating this and providing a GUI. I find that often when something (e.g. winetricks or installing some Windows requisites) doesn't work as expected in POL, I can also try navigating to POL's virtual drive and using Wine commands in the terminal.
On my Windows_Steam, I do not get the same error as the OP has in PlayonLinux, but much of the display is a black screen and unless I change it to "Small Mode" within the Steam settings.
I think what I've suggested here counts as a solution because it would give the OP the same functionality as they would have if they ran Steam via PlayOnLinux. It bears repeating again that the recommended approach is to start with the Linux version of Steam. But I haven't personally experienced any negative side-effects from approaching it backwards, and feel it has sometimes been a quicker way to get certain games running. And as a project, it perhaps gets us using some Wine commands directly without relying on PlayOnLinux's UI, and seeing how Steam puts the games it installs this way in analogous places to a native Windows install.
