Add attached filenames to subject in my applescript?
Simply change this line:
    set MailSubject to fileAliases

to this:
    set mailSubject to fileAliases as string

    tell application &quot;Finder&quot; to get insertion location
    set text item delimiters of AppleScript to result

    set mailSubject to rest of text items of mailSubject


ADDENDUM:
I hope you don't my observation that the script you present in your question contains a lot of redundant and/or badly formed code.  It works just fine, so it's not of great consequence beyond one of aesthetics.  However, a neater, well-formed script makes it easier to identify errors and—in the case of longer scripts that perform more tasks—will run more efficiently/faster.
So, with this in mind, I took the liberty to re-write your script, which I hope might be useful as a learning tool, or simply just be easier to get along with if you want to make further changes to it yourself.
    property mailSubjectPrefix : &quot;Datei(en) für dich: &quot;
    property subjAttNameSeparator : &quot;, &quot;
    property attachSeparator : &quot;*******&quot;
    property attachPrefix : &quot;Anhänge:&quot;

    set theSubject to &quot;Datei(en): &quot;
    set theContent to &quot;Datei(en) für dich:\n\n&quot;
    set recipientAddress to &quot;mail@mail.com&quot;


    -- Get the selection of the frontmost Finder window
    -- as a list of aliases
    tell application &quot;Finder&quot;
        set fileAliases to the selection as alias list
        set text item delimiters of AppleScript to insertion location
        set fileNames to rest of text items of (fileAliases as text)
    end tell


    -- Check if the selection is not empty
    if the number of fileAliases is 0 then return beep

    -- Now talk to mail to create the message
    tell application &quot;Mail&quot;
        set text item delimiters of AppleScript to subjAttNameSeparator
        tell (make new outgoing message at beginning ¬
            with properties {content:theContent, visible:true, subject:mailSubjectPrefix &amp; (fileNames as string)})
    
            -- Add the recipient
            make new to recipient at end with properties {address:recipientAddress}
    
            -- Attach all the selected files
            repeat with fileAlias in fileAliases
                make new attachment with properties {file name:fileAlias} at after the last paragraph
            end repeat
    
        end tell

        -- Put Mail in the foreground
        activate
    end tell

If you have any queries to need clarification on any points, feel free to leave a comment and I'll get back to you.