In Judaism, to be someone's son it doesn't necessarily mean to have the same DNA as the father, but it also means to have the same spiritual DNA, meaning to have the same way of thinking, of acting and of interpreting Torah. When Paul called Timothy his son, it doesn't mean Timothy was his son, but that Timothy was of the same spiritual DNA as Paul. Yeshua said "if you were Avraham's children, you would do Avraham's deeds", meaning to call yourself the son of someone implies to do the same deeds as the person you call 'father'.
All through the New Covenant, Yeshua is called 'Son of David' (Mathew 15:22, Mat. 20:31, etc), and that doesn't just mean He was from the davidic lineage, which He was, but it also means He had many characteristics as David.
There are many similarities between the two of them and I won't focus on all, but only on a few.
1. None of them had the right apparence to be Mashiach (Anointed). David was the last person Samuel would have picked to be anointed king and it's written about Yeshua 'when we saw him, there was no beauty that we should desire him.' Yeshayahu/ Isaiah 53:2b
2. They both had to leave Israel temporary and live among goyim (gentiles).
David had to leave Israel because Saul was chasing him, so he went in the land of the Philistines, and served king Akhish. (1 Samuel 27). While he was there he pretended he has nothing to do with Israel anymore and Akhish believed him as it's written in 1 Samuel 27:12 'Akhish believed David, saying, He has made his people Israel utterly to abhor him; therefore he shall be my servant forever.'
Goyim think the same about Yeshua, because they don't know The God of Israel, just like Akish didn't know David.
When David ran to Akish, Saul stopped chasing him and in the same way Jews have stopped searching for Yeshua because they know He's in the land of goyim.
Akish wanted David because he saw David was strong so he named David his keeper forever (1 Samuel 28:2).
Goyim act as if Yeshua is at their disposal, working for them, that they can make Yeshua do whatever they want just because temporary He is giving them grace.
Akish wanted to go to war against Israel so he summons David to come and protect him. -- Don't start a war against Israel and expect Yeshua to come and protect you.
David doesn't answer 'I won't go' but as a true Jew, he answers 'you shall see what your servant will do'. David went with his men but they "passed on in the rearward with Akhish' and when the Philistines see them they start yelling "What are these Hebrews doing here?"(1 Sam. 29:2) and they ask Akish to send the Jews back.
Goyim can't accept a Jewish Jesus. If somehow they see such a Jesus they make sure to send Him 'home' and they would rather believe that Yeshua is a Jesus with blond hair and blue eyes that was walking through Israel because He planned to let Himself be killed by Jews so He could blame them forever for it, and nothing else.
David couldn't fight against Israel although he couldn't live in Israel for that time, and Yeshua being ben David, meaning the Son of David, would never be against His own people, even though He is more known to the goyim, but doesn't mean He belongs in the lands of goyim, that's why when He'll return he won't live in Washington, nor London, nor Paris, nor Tokyo, but He will live in Yerushalayim (Jerusalem).
Not long after being sent back home, David returned to his home and became king over all Israel, bringing Aaron HaBrit (Ark of the Covenant) to Yerushalayim, ark that represented the Shekinah (glory) of Adonai.
It won't be long and Yeshua will return to His people Israel, He will return to Yerushalayim and He will bring HaShekinah there where it belongs. With Israel. And from there the whole earth will know that Adonai is One, His Name is One and He calls Himself The King of the Jews, The King of Israel.
Bat Melech בת מלך