"malware" by common definition is software of some sort that installs on your computer. The term is a conflation of the words "malicious" and "software" and hence clearly refers to programming code running without authorization on your computer with the intent of taking unauthorized control of your computer, stealing personal information, or performing some other act that would be harmful to the computer and/or its users.
A phishing email, since it is not software on your system but strictly a message that purports to be something it is not, would by the common understanding of the term not be considered malware. A phone call from someone claiming to be from your bank and asking for your bank account information would be the same, just via a different medium.
But this is strictly a argument over semantics and hence no point in continuing the discussion on that basis. If you wish to consider a phishing email to be "malware", you are free to do so, though you should understand that few other people will understand it on that basis.
The most important thing is that with a phishing email, unless it came with some sort of attachment or responding to the email authorizes download of something, there is nothing on your computer to remove other than the message itself. If you respond to such an email with personal information, then you have to address the problem with the service for which you divulged said personal data.
Regards.