I don’t have specific experience with TypeForm, but I do know that many platforms (including Integromat) have problems with Airtable’s single-select fields & multi-select fields — the exact problem that you’re describing. So it sounds like TypeForm has this same problem as well.
The easiest solution of all would be to just change your multi-select field in Airtable to include a new option called “Other”. Then, add a brand new single line text field in Airtable called “Other Response”, which would then appear on your TypeForm form when the user selected “Other” from the multi-select field. So, all new custom values would be added into this “Other Response” field in Airtable.
However, if you’d like to leave things setup as they are now, the easiest solution that I’ve found so far is to use Zapier as the middleman between TypeForm and Airtable.
Zapier has 100% full support for Airtable’s single-select and multi-select fields, so Zapier will be able to take the data from your TypeForm form and then put all of that data into Airtable — including multi-select & single-select fields.
(I bet that you can also do this with Airtable’s API, but that would be much more complicated to work with than just using an easy-to-use automation tool like Zapier.)
Hope this helps! If this answers your question, could you please mark this comment as the solution to your question? This will help other people who have a similar question.