Yes, vaccine induced psoriasis is real, though rare, flare ups of existing psoriasis also occur after viral illness, but is uncommon.
It's not unique to the covid vaccines, and AFAIK is not happening in unexpectedly high numbers.
The advice for those who have psoriasis is to get vaccinated., as the benefits significantly outweigh the risk of a rare reaction (and of course it could hapoen if you catch covid)
If the person is also in high dose immune suppressant medication, they should seek the advice of their hospital consultant to decide optimum timing for jabs