First job? Go with the financial firm, get 1-2 years of experience then leave. If you already had experience, I would say neither job is good. But getting your foot in the door is difficult. Most of your technical skills you pick up might not be transferable since it’s old tech like you mentioned but you should still get very valuable experience that you don’t have from school like working with version control strategies, release cycles, test/staging/production builds, writing tickets, prioritization, etc.
Some old tech is not as irrelevant as you may think too. If they have anything on Java, try to do that. It’s still widely used. But cobol, fortran, ada, php, that sort of thing… Those are dead or dying. You may not get the choice but you can at least state your preference and hope for the best.