My experience was the opposite: After starting T, I was better able to shake off stressors and handle adversity that would have stressed me out.
However, a lot of my experience was related to the process of HOW I started - taking hormone blocker first, then introducing a low dose of T, and taking weekly & not biweekly injections.
If your partner has JUST started T at a higher dose, in the biweekly format, and without stopping the E in the body first, yes, it could take time to sort itself out.
Also, some people experience the trough period (10-14 days after injection) much worse than other people and need to shift to a weekly schedule & some people even then are better off on daily/gel delivery for mood stability.
The best way for your partner to figure out how much of this is the T versus random life events is to keep a mood journal. Have your partner write down at the end of the day: (1) How stressed did he feel that day? (2) How well did he feel able to cope with that stresss? Scale of 1-10.
Do that for the first 2 months. Look for patterns. If it is all over the place, then it's not the T. If it is ALWAYS high, then your partner probably could benefit from counseling and/or stress management help. If it maps out in a pattern around injections OR around the timing of the menstrual cycle (if it still exists or where it would be if it was still happening), then you can identify more of the cause. And if it is related to the peak/trough of T, then your partner can adjust accordingly (more frequent delivery of lower dose).