Remove the current property reference from $_:

$hash = @{}

Switch -regex (Get-content -Path C:\Users\svats\Desktop\Fil*.txt)
{
    '^[a-z]+$' { 
        $name = $_
    } 

    '^\d{10}'  {
        $phone = $_
        $hash.Add($name, $phone)
        $name = $phone = $null
    }
    default {
        Write-Host "Nothing matched"
    }

}
$hash

Mathias R. Jessen's helpful answer explains your problem and offers an effective solution:
it is automatic variable $_ / $PSItem itself that contains the current input object (whatever its type is - what properties $_ / $PSItem has therefore depends on the input object's specific type).

Aside from that, there's potential for making the code both less verbose and more efficient:

# Initialize the output hashtable.
$hash = @{}

# Create the regex that will be used on each input file's content.
# (?...) sets options: i ... case-insensitive; m ...  ^ and $ match
# the beginning and end of every *line*.
$re = [regex] '(?im)^([a-z]+|\d{10})$'

# Loop over each input file's content (as a whole, thanks to -Raw).
Get-Content -Raw File*.txt | foreach {
  # Look for name and phone number.
  $matchColl = $re.Matches($_)
  if ($matchColl.Count -eq 2) { # Both found, add hashtable entry.
    $hash.Add($matchColl.Value[0], $matchColl.Value[1])
  } else {
    Write-Host "Nothing matched."
  }
}

# Output the resulting hashtable.
$hash

A note on the construction of the .NET [System.Text.RegularExpressions.Regex] object (or [regex] for short), [regex] '(?im)^([a-z]+|\d{10})$':

Embedding matching options IgnoreCase and Multiline as inline options i and m directly in the regex string ((?im) is convenient, in that it allows using simple cast syntax ([regex] ...) to construct the regular-expression .NET object.

However, this syntax may be obscure and, furthermore, not all matching options are available in inline form, so here's the more verbose, but easier-to-read equivalent:

$re = New-Object regex -ArgumentList '^([a-z]+|\d{10})$', 'IgnoreCase, Multiline'

Note that the two options must be specified comma-separated, as a single string, which PowerShell translates into the bit-OR-ed values of the corresponding enumeration values.

other solution, use convertfrom-string

 $template=@'
 {name*:Lori}
 {street:234 east 2nd street}
 {city:Raleigh nc 12345}
 {phone:9199617621}
 {mail:[email protected]}
 {name*:Robert}
 {street:2531 10th Avenue}
 {city:Seattle WA 93413}
 {phone:2068869421}
 {mail:[email protected]}
 {name*:Robert}
 {street:2531 Avenue}
 {city:Seattle WA 93413}
 {phone:2068869421}
 {mail:[email protected]}
 '@


 Get-Content -Path "c:\temp\file*.txt" | ConvertFrom-String -TemplateContent $template | select name, phone