This will fetch the SQL aliases from remote machines, using Invoke-Command, could make it into a cmdlet too...

Get-SQLAliasesRemote.ps1

#
# simple script that gets the SQL Alias info from remote machines 
#
[scriptblock]$GetSQLAliases = {

    $aliasRegistryPath = 'HKLM:\SOFTWARE\Microsoft\MSSQLServer\Client\ConnectTo'

    If (Test-Path($aliasRegistryPath)) {

        $aliases = @()
        (Get-Item $aliasRegistryPath).Property | ForEach-Object {
            $aliasName = $_
            $aliasTarget = Get-ItemPropertyValue $aliasRegistryPath -Name $aliasName
    
            $aliasTargetType = switch ($aliasTarget.split(',')[0]) {
                'DBNMPNTW' { 'NP' }
                'DBMSSOCN' { 'TCP' }
                Default { 'UNKNOWN' }
            }

            $aliasTargetServer = $aliasTarget.split(',')[1]
            $aliasTargetPort = $aliasTarget.split(',')[2]

            $aliases += [PSCustomObject][Ordered]@{
                'AliasName' = $aliasName
                'Protocol'  = $aliasTargetType
                'Server'    = $aliasTargetServer
                'Port'      = $aliasTargetPort
            }  
        }
        return $aliases
    }
    # if the registry key doesn't exist there are no aliases configured
    else { return $null }
}

# example usage:
$aliases = Invoke-Command -ScriptBlock $GetSQLAliases -ComputerName LABSQL01, LABSQL02
# Invoke-Command will return the computer as "PSComputerName" - can rename to "ComputerName" using an expression, e.g:
$aliases | Select-Object @{name = 'ComputerName'; expression = 'PSComputerName' }, AliasName, Protocol, Server, Port | Format-Table -AutoSize

<#
Example output:

ComputerName AliasName      Protocol Server                    Port
------------ ---------      -------- ------                    ----
LABSQL01     myalias        TCP      labsql02                      
LABSQL01     my alias2      TCP      labsql02                  5353
LABSQL01     anamedpipe     NP       \\labsql01\PIPE\sql\query     
LABSQL02     labsql02 alias TCP      anotherserver             1567

#>