There are several external solutions for that:
https://boxshot.com/ai-toolbox/tutorials/dynamic-texts/
https://www.esko.com/en/products/dynamic-content
And https://3fl.jp/is051/, in some cases

I need this feature also,

When making some technical drawing, it's really boring to enter a text which should be the document name...
without all the mispelling you can do...

Is it possible to add a similar feature to "text variables" from InDesign to Illustrator? In particular - the file name, modification date.
It would be even better if there were some variables, for example - author, size of the canvas, minimum size of the text used in the project.

Meanwhile you can use Batch Text Edit script or native Find and Replace.

This can be done with Variables manually, somewhat easier with the Variable Importer script, but the method generally sucks.

Limitations of it are described here (and Vasily Hall, the author of the script, comments there):
https://illustrator.uservoice.com/forums/601447-illustrator-desktop-bugs/suggestions/36934588-data-merge-with-variables-shared-by-multiple-objec

Text can function as a smart object. Select two text boxes that have the same text (ex. "July 2021") and apply a feature that pairs the two, so when one is updated, the other updates automatically as well.

Currently there is no 'automatic' way of entering repeatable variables into text fields. For example a date at the top of each page, or an incremental page number, or a formula etc. Can Illustrator please add maybe a custom field onto the existing variables panel?

The main benefit would be for labelling or repeating document information across pages or in multiple places within a document.
Example. I Create a header page with a date "January 2020" and a version number 'Ver 23'. I would add these two values to a custom variable palette and can then create linked text fields at will across my document(s) without having to retype them every time. If I want to update the version or Date, I only have to type the new value once and ALL the linked text fields will automatically update.

Yes, this is similar to how pages and master data work in inDesign, but just a little of this feature would be useful here.

This is almost similar to having a library item, such as an external logo, but this would be stored internally with each document.

Even better when inputting text, you could press your custom entry on the dialog window and it would enter it into the text. I see the format of dates, and output to also be customizable.

Variable, editable, movable and positionable, save always upon save, date time, user, file path, etc, will be great future...

Have you tried the VariableImporter.jsx script?
https://github.com/Silly-V/Adobe-Illustrator/blob/master/Variable%20Importer/VariableImporter.jsx

It would be really helpful to have an option in Illustrator which can automatically fill custom variable fields like date, file name, etc.

I am working in print industry for almost 10 years and our primary software is AI, with adding the above option, it will help designers to save lot of time.

Appreciate your consideration for the above idea.