No built-in predefined variables or other things to get the "build name" you want. If you intercept the network traffic analysis, you will also find that the "build name" you want is not a whole, it is a combination of two data.
You need to design your own code to get it. For example, if you are based on Azure Git Repo, below pipeline YAML can help you get the "build name" you want when creating a pull request to trigger the pipeline:
trigger:
- none
# 1
stages:
- stage: s1
displayName: Get the lastest comment
jobs:
- job: testJob
steps:
- task: PowerShell@2
name: setvar
inputs:
targetType: 'inline'
script: |
$PAT = "<Your Personal Access Token>"
$org_name = "<Organization Name>"
$project_name = "<Project Name>"
$branch_name = "<Current Branch Name>"
$def_id = "<Pipeline definition ID>"
$run_id = "$(Build.BuildId)"
Write-Host $run_id
if("$(Build.Reason)" -eq "PullRequest") {
$headers_run = New-Object "System.Collections.Generic.Dictionary[[String],[String]]"
$headers_run.Add("Authorization", "Basic "+$PAT)
$run_url = "https://dev.azure.com/"+$org_name+"/"+$project_name+"/_apis/pipelines/"+$def_id+"/runs/"+$run_id+"?api-version=6.0-preview.1"
$response_run = Invoke-RestMethod $run_url -Method 'GET' -Headers $headers_run
$response_run | ConvertTo-Json
$pull_request_id = $response_run.variables."system.pullRequest.pullRequestId".value
$headers_pr = New-Object "System.Collections.Generic.Dictionary[[String],[String]]"
$headers_pr.Add("Authorization", "Basic "+$PAT)
$pr_url = "https://dev.azure.com/"+$org_name+"/"+$project_name+"/_apis/git/repositories/ShowBuildName/pullrequests/"+$pull_request_id+"?api-version=6.0"
$response_pr = Invoke-RestMethod $pr_url -Method 'GET' -Headers $headers_pr
$response_pr | ConvertTo-Json
Write-Host $response_pr.title
$str = $response_pr.title
Write-Host "##vso[task.setvariable variable=outputvars;isOutput=true]$str"
}
else {
$headers = New-Object "System.Collections.Generic.Dictionary[[String],[String]]"
$headers.Add("Authorization", "Basic "+$PAT)
$url = "https://dev.azure.com/"+$org_name+"/"+$project_name+"/_apis/git/repositories/ShowBuildName/commits?searchCriteria.itemVersion.version="+$branch_name+"&api-version=6.0"
$response = Invoke-RestMethod $url -Method 'GET' -Headers $headers
$response | ConvertTo-Json
$str = $response.value[0].comment
Write-Host "##vso[task.setvariable variable=outputvars;isOutput=true]$str"
}
# 2
- stage: s2
displayName: Get the "build name"
dependsOn: s1
variables:
vars: $[ stageDependencies.s1.testJob.outputs['setvar.outputvars'] ]
jobs:
- job:
steps:
- task: PowerShell@2
inputs:
targetType: 'inline'
script: |
Write-Host "$(Build.BuildNumber) $(vars)"
Pipeline run:

Successfully get the "build name":
