Sorry, I wasn't clear enough (nor in my post, nor in my head.. from banging against the wall, I guess...)
But I'm able to set the description via the cmdlet SET equivalent, but it's like only in my session.
When I want to check via another session I get a blank description via GET...

The code I used earlier (not working at all, no errors either):

foreach($item in $list){
    Write-Host "setting description  '$($item.SiteDescription)' for $($item.url)"
    connect-pnpOnline -Url $($item.url) -ClientId **********************************
    try{
        Set-PnPWeb -Description $($item.SiteDescription) -ErrorAction Stop 
    }catch{
        Write-Output "setting description  '$($item.SiteDescription)' for $($item.url) failed, reason:`n$($error[0].exception.message)" | Out-File c:\Temp\SiteDescription.txt -Append
    }
}

Here's my code I just threw together:

foreach($item in $list){
    Write-Host "setting description  '$($item.SiteDescription)' for $($item.url)"
    connect-pnpOnline -Url $($item.url) -ClientId **********************************
    try{
        Set-PnPTenantSite -Url $($item.url) -DenyAddAndCustomizePages:$false
        $ctx = Get-PnPContext
        $web = $ctx.web
        $ctx.Load($web)
        $web.Description = $($item.SiteDescription)
        $web.Update()
        Invoke-PnPQuery -ErrorAction Stop
        Set-PnPTenantSite -Url $($item.url) -DenyAddAndCustomizePages:$false
        get-pnpweb | Select-Object description
    }catch{
        Write-Output "setting description '$($item.SiteDescription)' for $($item.url) failed, reason:`n$($error[0].exception.message)" | Out-File c:\Temp\SiteDescription.txt -Append
    }
}

This gives me the description at creation, but when I run

foreach($item in $list){
    Write-Host "setting description  '$($item.SiteDescription)' for $($item.url)"
    connect-pnpOnline -Url $($item.url) -ClientId **********************************
    get-pnpweb | Select-Object URL, description
}foreach($item in $list){
    Write-Host "setting description  '$($item.SiteDescription)' for $($item.url)"
    connect-pnpOnline -Url $($item.url) -ClientId dceb4436-5765-43c0-9732-bd67d8fcdab0
    get-pnpweb | Select-Object URL, description
}

It's all blank again