The use of AI and voice recognition seems mostly designed to make the result seem more sensational than it actually is. Does any computation actually happen in the "organoid" part? How would you even train such a cell to perform a task?

From reading the article it seems to me that the answer is no. The actual contribution is feeding the organoid electric signals, and reading its reactions. (Probably the machine learning algorithm used would have had even better accuracy, if the input signal hadn't been fed through a layer of goo. It doesn't say whether this is the case.) The rest is speculation of future applications.

> To test Brainoware’s capabilities, the team used the technique to do voice recognition by training the system on 240 recordings of eight people speaking. The organoid generated a different pattern of neural activity in response to each voice. The AI learned to interpret these responses to identify the speaker, with an accuracy of 78%.

It "generated a different pattern," with no indication that this pattern was optimized to be useful in any way.

I think the key part of a (bio-)"computer" is the possibility of programming/training it, not just reading input from it.