First of all they both clearly have red hair, but they also have a scar on their left eyes (althought this similarity does not make much of a difference, it is there to show the similarities.) Now they both look fairly alike.
Now for Shanks being Kid's dad:
Shanks is a yonkou and became a pirate far before this current new generation ("the worst generation"). I believe after Buggy and Shanks were on Roger's ship, Buggy left clearly before Shanks. But Shanks clearly said that he wants to become a pirate for freedom in the sea (and things like that) unlike Buggy for the gold and fortune. Shanks must have left most likely shortly after Buggy left, feeling it is his time. I believe Shanks would have grown up and may have had a child (Roger had a child without anyone knowing and it seems reasonable to assume Shanks would do the same). Shanks would then leave the child with the mother. Kid was born in South Blue and Shanks from West Blue. It is obvious that Shanks travels and he may have found his true love at one point and had a child with her. Shanks may have left the baby with the mother to continue with his search for freedom and continue being a pirate similar to Dragon leaving Luffy at a very young age and like Ace. Ace grew up to hate his father, Gol D Roger because he was never there for him. Kid would feel the same way after finding out his father, Shanks left him to continue being a pirate and leaving his mother behind. There is the possibility that Kid's mother also died later which angered Shanks for not being there. This could have resulted in Kid's aggressive and angry attitude (especially his first appearance after the timeskip).
Kid's Revenge:
Kid knew Shanks became a notorious pirate, so he grew up like Ace with hatred towards his father and wanted to become a pirate just like Shanks, except better, and that is why Kid wants to find the one piece, so he will become pirate king (most likely kill Shanks and become a better pirate than Shanks ever was). It may be Kid's dream to kill Shanks and beat him for what he left him for. Kid would have this in mind growing up to accomplish his goal. Shanks most likely carefree as always may not have ever noticed or remember Kid and would have forgotten him or possibly went back to find him, but in no luck as he already left to become a better pirate than Shanks and finds out his wife died. Shanks would assume Kid is dead too. Kid is using his mother's name as Eustass. Shanks is a good guy and would go back to see if his child is doing well. He never wanted him to be angry at him.
Luffy and Kid Connection:
Shanks absolutely adores Luffy, and gave him his signature strawhat.
Assuming his real son would be dead, he became a father figure for Luffy that is a lot similar to Kid in that they both had a father leave them at a young age, and is like an orphan. Shanks sees Luffy as the son he wished he had, so wanted Luffy to become the pirate king. Kid would clearly be upset and outraged that Shanks gave his signature hat to a random kid he met and not his son, Kid. Kid is clearly angry at Shanks, but Shanks has no idea or even that fact he has this son.
Luffy, Law, and Kid Meet:
When they first meet and fight a lot of marines together, (5 supernovas in one place). In this fight, Kid clearly had a hatred towards Luffy and Law (but mostly Luffy) especially when Luffys hat flew off Kid may have felt that should be his. Kid did not like Luffy fighting with him as he is his enemy and like Shanks' "new son". There was also one part during the warford arc in which Kid saw that Luffy was doing things in the war and got angry at the newspaper not liking Luffy getting attention (Kid wants that attention from the world and shanks).
Timeskip:
Kid became stronger but still seeks the time to get revenge.
Now, Luffy and Law are in an alliance, but Kid, Hawkins, and Apoo are in another alliance simply to not be in Luffy's alliance and gather some attention to himself too. Law and Luffy are after Kaido, but Kid's alliance is rumored to go after Shanks. This is what Kid intended from the start to kill Shanks and become a better pirate than him and kill him (he would be the new yonkou and take his spot). Kid would get his revenge, but his revenge would not end until he also beats Luffy. By beating Shanks, Luffy would be outraged and Kid would be satisfied to see that Luffy is suffering the same way he felt when Shanks left him and gave his hat to Luffy. Luffy may then understand Kid's scenario. Kid would then want to kill Luffy to get his revenge as well. Kid is angry because Shanks forgot him and found a "a new son" and gave his hat to him instead of his own kid.
