Yes a man can catch some forms of bacterial infections from a woman but not all types of bacterial infections a woman has can be transmitted to a man.
For example the bacterial infection BV cannot be transmitted to the man because a man cannot catch the BV bacterial infection from women.
Women can catch BV though from another woman through sexual activity with that woman though.
But also a man who has BV can sometimes spread the BV through sexual intercourse to a woman.
But yes you can as a man catch some bacterial infections from a woman through sex and sexual contact and woman can sometimes catch some bacterial infections from men through sex and sexual activity.