Firstly, PowerShell does not store any PIDs anywhere. That's not a job for a scripting language, at all. It accesses them through Windows .NET APIs and they are of course only stored on the system/kernel level. They are only assigned as part of temporary objects when using certain functions, such as return value of Start-Process.
Secondly, keep in mind that processes can be very dynamic - they can quickly appear and disappear. The chrome.exe executable you're starting is just a launcher and/or process manager for Chrome. The window itself is a different process - each of windows has a separate one. Even each non-frozen tab can have own process.
Assuming you only run one window, the shortest way to kill it in PowerShell would be something like:
Get-Process chrome | Stop-Process -Force
as the -Name param is optional and this matches all chrome.exes exactly.
Or even simplier:
Stop-Process -Name chrome -Force
Otherwise you need to match it by e.g. window title.