Is there a general form and form data handling mechanism in WordPress that I can facilitate for this, like e.g. WordPress has a form object containing a “field” or data object for each of the form’s fields, allowing me to access the form(s) on a page?
No, there is not.
I actually wouldn’t even need a form object. Instead, I could just place a couple of input fields on the page. How would I access these in some php function?
The same way any other PHP/HTML application would. WordPress does not provide a form API.
How would I access the forms (is there e.g. a WordPress function receiving the form name and returning me a form object?)
How would I intercept loading of the page containing the form(s) (or any page, that is) before it gets rendered, but after the form object is instantiated/initialized?
You don’t, there are no form APIs or objects in WordPress, that is not a feature WordPress implements.
As I said, I am not looking for a particular form implementation by e.g. some plugin, but whether there is a general underlying mechanism, which might eventually be utilized by such plugins, I could facilitate.
No such mechanism exists. Each form plugin builds their form system from scratch. Some use custom tables to store the data, some use custom post types, but there is no generic system that sits underneath them. You would have to build it yourself or use a 3rd party plugin/library to provide it.