late to the party, catching this on Amazong Prime

It def adds to the overall story if you watch his Doctor Who episodes. I don't remember exactly how/why, but I got started on Torchwood first and wondered wtf the deal was myself, which led me to Doctor Who, and I'm glad that it did. It won't make as much (if any) sense why he's always nostalgically referencing "...the right Doctor" unless you invest in both shows. Hope that helps and if you do invest the time (Capt. Jack appears midway through 'Series' 1 (2005), start off with the first episode "Rose" and If you stick with it through the end, you won't be disappointed. I love DW, but a lot of the first season can be skipped over to allow focus on Jack's character, but bare bones, I'd recommend Eps 1, 2, 8 (so you know what's going on & they're damned good stories), then 9 (Jack's first appearance) through 13 - the finale is pretty essential to understanding/appreciating his character. Enjoy, homie!

There is a big hent to the possibility of who Jack is at the end of I think the 3rd or 4th series finale of Docotor Who. It's not definitive and its left purposely ambiguous.