Is it permissible to secretly recite during a missed Dhuhr prayer if another person is praying in congregation with the intention of Isha, and are both of their prayers valid in this case?

The answer can be summarized in two points:

First: The ruling on reciting aloud or silently in a missed (qaḍāʾ) prayer: Many scholars hold the view that a missed prayer should be performed in the same manner as it was missed, whether it was to be recited aloud or silently. This is the view preferred by the two Sheikhs, Ibn Bāz and Ibn ʿUthaymīn.

Second: The permissibility of someone performing a missed prayer (qaḍāʾ) being led by someone performing a current prayer (adāʾ), or vice-versa: The majority of scholars held that this is not permissible. However, the Shāfiʿīs permitted its validity, and this is the preferred view. The difference in intention between the follower and the imam does not matter, as the prohibition of discrepancy relates to outward actions, not inner intentions.

Therefore, your silent recitation in the missed prayer is valid and consistent with the majority view, and the validity of the prayer of someone who followed you is consistent with the Shāfiʿī school of thought, which is the preferred view.