From what I understand, if a Jewish person wants to leave Iran, they can't take their money with them. So all the very poor Jews left since this wasn't a concern, and all the very rich Jews (who had ways of moving money) also left. Those that are left are the middle class Jews who have comfortable lives and fear the poverty they may face if they leave.
They are free to travel abroad but not to Israel, (where many have relatives). I met someone who had to flee Iran because they visited Israel via a second country and got found out.
The Iranian Jews I know are very proudly Persian. Their parents or grandparents might have left the country 50 years ago but they strongly identify as Persian Jews, have their own communities and customs and usually marry each other.
@Vinvertebrate I have heard that about anti-regime Iranians being very pro-Israel. It doesn't surprise me.