Grammatically, there is no difference between "write me" and "type me" but tradition requires that they be treated differently. The reason for this is found in the long history of letter writing. Writing letters used to be the main way people communicated. It used to be that in order to say something to someone, you needed to send them a letter. For that reason "write" behaves a lot like "send" or "throw": "write me" "send me" "throw me" "write a letter to me" "send a letter to me" "throw a ball to me" "Type" follows a different tradition. In our minds, it is not connected to the idea of sending. It is only connected to a mechanical process of punching keys on a keyboard. If you said "type me", I would be confused and ask you "what do you want me to type?". If you said "type me an email", I would understand what that means, but typing is a mechanical chore so it would sound less friendly than "write me an email".

If someone said "write me" in an American accent, I would assume they meant "write to me". A British person of my age would be very unlikely to say "write me" (that's an Americanism) and unlikely to just say "write to me" (although that is fine as British English) but might say a fuller sentence such as "please write me a report" or "please write me a letter" or "please write to me when you get there". Gen Z may have copied more Americanisms from the Internet etc. If someone said "type me" (and nothing more) I would probably be rather puzzled as to what they meant. "Type me a letter" makes sense but may sound a little odd (a handwritten letter won't do?), and "type me an e-mail" likewise makes sense but may sound a little odd (e-mails are always typed... why not just say "send me an e-mail"?). Depending on context, "type me a message" could be a fairly natural thing to say I guess.

In addition to what Mila said, you could say 'write to me,' or 'write me an email.'

While someone will definitely understand what you mean if you say that it is not common to say "type me". If you want someone to text you , you could say "Text me" . or "Send (me) a text" If you want someone to send you an email it's better to say" Send (me) an email " or "email me""