Obviously the tool to which you refer is native - and has absolutely nothing to do with Adobe products and Apps. So we can dismiss discussion of Adobe in its entirely…
Insofar as I’ve been able to determine, the Add Text Form feature that you see when viewing a PDF document from the Files App, is not yet documented within the iPad User Guide - and neither does there appear to be a supplemental Apple Support page that describes the feature.
However, experimentation demonstrates that the feature does pretty much as its title might suggest, in that the tool will insert an editable Text Form within a PDF page.
With this tool active in markup tools, tapping the blue “+” button (at bottom right) will drop a Text Form Box on the visible page - which can then be dragged to where you wish to place the text box. Once positioned, you can adjust the width of the box with the two drag-handles with which you should already be familiar. As long as the tool remains active, all inserted Text Form boxes are visibly highlighted - and are editable.
To insert or edit text with a Text Form box, simply tap the box to expose the editing menu; tap Edit to enter text within the box. You can now enter text (the default text style appears be Times New Roman), using the carriage return key to add an additional line, the Text Form Box expanding as additional lines are added. Tapping the screen anywhere outside the box ends text entry. You can continue editing by again tapping the box and selecting Edit from the menu.
At first this tools appears to limited in its capabilities, although there is more to explore…
If you tap the Markup button to expose the Markup Toolbar before enabling the Text Form Tool, the visible Markup Toolbar will change to show only those tools that are applicable - such as Text Style (Helvetica, Georgia, Noteworthy), text size slider and text colour-picker.
You should note that you may need to dismiss the keyboard to see the Markup toolbar to make selections. The UI for this feature is not particularly elegant - and requires a dance between selection buttons to select tools and edit the inserted Text Form boxes - but with practice is usable.
In summary, this feature might appear to be “work in progress”. Perhaps the feature will be refined over time - and eventually be documented by Apple.