As suggested by Santiago Squarzon, simply pipe into Group-Object:

Get-Content "c:\listip.txt" | 
    Select-String -Pattern '^(?:(?:25[0-5]|2[0-4][0-9]|[01]?[0-9][0-9]?)\.){3}(?:25[0-5]|2[0-4][0-9]|[01]?[0-9][0-9]?)$' | 
    Group-Object -NoElement | 
    Select-Object Count, @{ n='IP'; e="Name" }

Output:

Count IP
----- --
    4 1.1.1.1
    9 8.8.8.8

The parameter -NoElement removes the Group property that is output by default by Group-Object, but is not needed for counting only.

I’ve added Select-Object to rename the Name property to IP, using a calculated property .

If you’d like to correctly sort the unique IPs (i. e. not string sort, but number by number), you might look at this answer.