Hcfyk

This sounds like Google Docs, perhaps combined with Trello (for the Kanban board and calendar view), would do for you.

It's got live updating -- I don't think it tracks edits by author exactly, so you could agree on a convention -- like my students on one team devised a rule where each of them owned 2 colors. (4 students, so like light blue and dark blue was one, red and pink was another, etc.) Anything NEW they would do in one of their colors. Any EDITS to others' work, they'd do in their edit-color. (They copied original sentence to comments, in case people wanted to compare.) They would comment each paragraph with "ok" if they had no changes. Their teamleader would turn paragraphs BLACK if everyone had said "ok".

Since it's only 2 of you, that part could be simpler.

If you want to stay in Google things, you could fake a KanBan with Google Sheets, just name your columns, and each spreadsheet Cell becomes an Item, which you can move through the process. And then Google Calendar of course also exists for deadlines or blocking out time to write.

But Trello is free to use and lets you link to Gdocs also, so that's probably a better Kanban/Deadline solution.

Only tool I can think of would be Google docs.

You can have multiple people writing in a document at the same time and see where their cursor is, there is no color coding on who's typing by default.

You can enter what's called "suggestion mode" where changes you add are not validated immediately but merely appear highlighted to your color, allowing the owner to accept or refuse the change.

For updates, it keeps track of any changes through the document, so any day you can view the history of the doc and it will show you what was changed and by whom.

There is however no task or timeline managing that I can think of.

Another alternative (if you are not afraid to use LaTeX) would be Overleaf. While a learning curve exists you can layout documents beautifully with it.

I come from an engineering background where special characters and formulas are needed on a regular basis. We used Overleaf within our team to work on several parts of our document simultaneously.

Not for simultaneous work on one document, but maybe of interest for you: should you get off the grid for some reason, you can use offline editors like TeXstudio to continue working on your files. Once you get back online, you would still have to carefully merge stuff together, but there won't be many bad surprises in regards to formatting.

EDIT: I missed some of your points, sorry.

You can see where your colleague is working (there is a visible cursor) and who is logged in atm. I do not recall any other highlighting, though.

You can leave comments within the text visible for everyone working on the document - this might offer you the task management functionality without overburdening you with yet another tool.

As for deadlines... I did not notice any tracking functionality in Overleaf. I would personally use Gantt charts to organize myself, including milestones and the like. GanttProject offers everything I need, only used it offline though. But if you put the project files into some secure cloud storage or inside a git repository, it may suit your needs.

EDIT2: As Eric Lino pointed out in the comments, there can only be two people working on the document if you are using the free version (source: pricing list).

Depending what you mean by 'online', there's Gobby

https://gobby.github.io/

I want to mention Confluence Wiki ( https://www.atlassian.com/software/confluence )
as it supports editing the same Wiki page where you see the cursor of the other Editors as well what they are typing.
For less than 10 users the licence is 10$ single payment (which gets donated) - and can be setup on your own server.
You could create one Wiki page per chapter for example. - The structure (as on wikis in general) is your choice.
There is also a version history to see what changed from version x to y

Depending on what you use for writing, you should consider git. With LaTeX it works like a charm, each of you can create his own branch and merge when he finishes a chapter/ paragraph. Git will keep track automatically about who made which changes. And with git diff every change will be highlighted for you.