A word like "bid" in increasing manner but used metaphorically

Your two examples are slightly different, and different phrases are used:

Everyone is trying to get on the game of who suffers more, then everyone is (that verb).

upping the ante.

OED:

Ante: (n.) 1. In poker and similar card games. a. A mandatory stake put into the pot or betting pool before play begins

to up the ante: (v.) figurative. : [...] (b) (in extended use) to increase what is at stake or under discussion.

1986 Telegraph (Sydney) (Nexis) 5 Nov. Mr Cain said he did not wish to make ‘gratuitous threats’ or ‘up the ante’ during the dispute [with nurses].

Alice: i bought a PlayStation 4. Bob: i bought a PlayStation 5. Alice: why the fuck do you always (that verb)

go one better /top everyone's story.

P5 (a) to go one better: to (narrowly) outdo someone or surpass a previous effort or achievement.

2007 B. Warner Sit down & shut up xxi. 209 When just having short hair didn't seem radical enough, the hardcore punks went one better and shaved it all off.

to top IV. 12.b. To surpass, excel, outdo; to cap.

a1616 W. Shakespeare Coriolanus (1623) ii. i. 20 Topping all others in boasting.