The Arsenal winger was named man of the match in England’s final group-stage game and was also impressive in spells after keeping his place for the last-16 win over Germany.
Saka will now be hoping to stay in the team for Saturday’s quarter-final clash with Ukraine in Rome.
But for Shaw, the 19-year-old has also played a vital role in keeping the team smiling off the pitch as they prepare for matches at their St George’s Park base.
“Honestly, I love him,” Shaw told the Lion’s Den.
“I would just love it if he was my child or something, I love him like that and you could ask anyone in the squad and they would say how nice a guy he honestly is.
I didn’t know him before I came to the camp and I didn’t know what he would be like but I have got on so well with him, I would love him to be my brother.
“He is so cool, so funny and he makes everyone laugh and he doesn’t mean it.
“He is not a loud person it is just the way he is, the way he speaks. Someone like that in the camp is good to have and he gets everyone laughing and we just all love him and appreciate the way he is.”
Shaw was also full of praise for Saka’s on-pitch displays with the pair having both started the last two matches.
Asked about Saka’s performances, the Manchester United full-back replied: “He is unbelievable isn’t he?
“He is still only 19 that is the scary thing, he takes everything in his stride, he is not fazed by anything.
“When he came in (to the team) he has been brilliant so far and I keep telling him to keep it up and he keeps impressing everyone.”
Shaw revealed he and Saka are just two members of the squad who are now watching dating show Love Island on a daily basis.
“Everyone (is watching) – I think most of the squad,” he added.
“We have got a big screen where we watch the game and sometimes we watch a movie together.
“But now, in the evenings it is Love Island. Nine o’clock. That is it.
“You book your treatment time so you are just on the bed when Love Island comes on and most of us just sit there and watch it. It is something to watch, it is good and gets everyone talking.”
While not tuned into Love Island, Shaw may be forgiven for casting an eye back to the 2-0 win over Germany at Wembley on Tuesday evening.
The 25-year-old went into the game having laughed off the latest criticism from his former United boss Jose Mourinho and played a hand in both goals, including laying on the cross for Raheem Sterling to break the deadlock.
“Everything about the last day or two has been unbelievable, I swear to God I have not felt this happy in a long while and I hope the fans feel the same way,” he added.
“The experience has been amazing, the spirit of the squad and the togetherness is something I have not experienced before.
“Everyone is so close and gets on so well and the way they have made up St George’s Park has been great.
“I can’t describe it, the game speaks for itself really, we tried not to big the game up but I think everyone knew how big the occasion was – how much it meant to us as players and of course the fans.”