In the opening of season 2 of One Punch Man, there is a shot where you see Saitama with his friends (Genos, King, Bang , Fubuki) and also Metal Bat and Tatsumaki. Is there any obvious reason why Metal Bat and Tatsumaki are there? Do Metal Bat and Tatsumaki get any closer to Saitama at any point of the story in the manga?
For Metal Bat, they don't have any significant interactions so I doubt they'd get closer.
For Tatsumaki, I'm not sure if this can be called getting closer but they
fought during the Psychic Sisters Arc
(this is canon as the webcomic is where the manga is based from). Afterwards,
Saitama displayed concern and interest toward Tatsumaki, as he ran after her to see if she would be alright flying home with her injuries...Saitama then asked her why she became a hero if she's trying to cut relationships with people.
At this point, Tatsumaki almost revealed her past to Saitama but didn't. Instead a flashback was shown when she first met Blast, which, to me, reveals Blast's true identity. To quote from the wiki,
She witnessed a man defeating a giant multi-eyed monster. His name is Blast. His appearance has a strong resemblance to Saitama. In fact, Blast said to Tatsumaki that he is a hero for fun, echoing Saitama's words. Blast ends his conversation by telling Tatsumaki that she has to be strong, because not everyone can save her. This motivated Tatsumaki to become a hero, and this is also a reason why she wants to be alone and cut relationships with people.