How save specific strings and then put next to lines?

Perhaps the following can help you get your desired final output, it is a more manual and classic approach, it doesn't intent to solve everything with regex but it does help with the logical conditions:

Get-Content file.txt | ForEach-Object {
    # if the line starts with CODE= followed by a code
    if ($ -match '^CODE=(.+)') {
        # capture the code (e.g. MMH4, PPH2, etc)
        $code = $Matches[1]
        # and go to next line
        return
    }
    # if we have captured a code in previous lines
    if ($code) {
        # loop thru each match of a line containing a combination of ABC=...&&DEF=...
        # we use [Regex]::Matches(..) here as there could be multiple || conditions
        foreach ($match in [regex]::Matches($, '(?:ABC=\d+&&DEF=\d+)+')) {
            # replace && with a TAB concatenated with TAB and the captured code
            "$($match.Value.Replace('&&', "t"))t$code"
        }
        # then we clear this variable to look for a new line starting with CODE=
        # NOTE: See the end of the answer to understand the usage of this
        $code = $null
    }
}

The above, using your sample text would produce the following output:

ABC=704 DEF=03  PPH2
ABC=706 DEF=04  PPH2
ABC=334 DEF=030 KK7

You might notice the use of $code = $null, this is a tiny optimization to not enter the if ($code) condition more than once for each appearance of a line starting with CODE=, it could be removed and removing it could have a different effect depending on the input, taking this as a small sample:

CODE=KK7
description=beg Area4 - ABC=334&&DEF=030
description=beg Area4 - ABC=335&&DEF=031
END

If left as-is, the result for the sample would be:

ABC=334 DEF=030 KK7

Whereas if that line of code is removed, the logic would match both inner lines:

ABC=334 DEF=030 KK7
ABC=335 DEF=031 KK7

---

EDIT

Looking at the latest edit on your question, you're seemingly wanting to convert your text into a TSV, for this I'd recommend to use a function where you can output the headers in its begin block; you could technically use ForEach-Object -Begin {...} too if you like.

So the code doesn't change that much from what was provided above, the regex pattern will use 2 capturing groups to get the value of ABC= and DEF=... see here for details: https://regex101.com/r/qvGBZy/1.

function ConvertTo-Tsv {
    param(
        [Parameter(ValueFromPipeline, Mandatory)]
        [string] $Line,
        # using a parameter instead of hardcoding them in begin {..}
        # gives more flexibility, you can provide new headers when needed
        [Parameter(Position = 0)]
        [string[]] $Header = ('ABC', 'DEF', 'CODE')
    )
    begin {
        # output the header joined by TAB
        $Header -join "`t"
        $re = [regex]::new('(?:ABC=(?\d+)&&DEF=(?\d+))+')
    }
    process {
        if ($Line -match '^CODE=(.+)') {
            $code = $Matches[1]
            return
        }
        if ($code) {
            foreach ($match in $re.Matches($Line)) {
                # output values of ABC= and DEF= with the value of CODE=
                "$($match.Groups['abc'])t$($match.Groups['def'])t$code"
            }
            $code = $null
        }
    }
}

Then the usage is pretty straight forward, read the file content, pipe to our function and lastly pipe to file storage:

Get-Content myFile.txt | ConvertTo-Tsv | Set-Content myOutput.tsv

The expected output using the function given the sample input text would be:

ABC     DEF     CODE
704     03      PPH2
706     04      PPH2
334     030     KK7

And we can tell this is a valid TSV given that a conversion from it produces the following objects:

PS> Get-Content sample.txt | ConvertTo-Tsv | ConvertFrom-Csv -Delimiter "`t"
ABC DEF CODE
--- --- ----
704 03  PPH2
706 04  PPH2
334 030 KK7