The peak input is 2,1 A, your power bank must output that reliably in the first place, apart from that it doesn't matter how you give it 5 V (while the specs are plausible, the product seems fishy as it's explicitly advertised "for arduino", which without MANY accessories doesn't need anywhere near 1 A... and it would run more efficiently off USB power in the first place, lol)
Of course, I hope you know if the plug and voltage are right for the PS1 slim, because I don't have one!
This is an application where designing your own power bank (or at least combining reasonable selections of charge controller, 2S/3S lithium battery, switching converter) would pay off in efficiency (heat and battery life), though I understand it's not easy for the complete beginner
