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The custom form is posting to their server api.qubole.com, and then their server code (could be Java, PHP, C#, Ruby, whatever) is remapping fields to the Marketo equivalents and reposting to Marketo. For example, the form gathers the field user[email] but the real Marketo field is Email, so their code is not simply passing form posts through to Marketo but changing them in important ways.
To make matters much worse, they aren't even running Munchkin, so Marketo can't associate the form post with a known lead.
Because they are reposting to Marketo's forms endpoint (as a real form would) you still see a Filled Out Form activity for a form with a certain ID + name, but as you noticed you have no control over what is presented on the form. There's no relationship between the Marketo form descriptor and the JS they use to control the form in the browser, nor any connection to the other code they use on their server.
This is not a recommended architecture for a number of reasons, not only because of the above problems but because they are limited to 30 form posts total per minute (meaning a hacker could bring down the form functionality easily).
Thank you, Sanford! Much appreciated!