When Boimler is beamed down to the planet, he is returning Spock's Vulcan salute with one hand while holding the metal object in the other. When he materializes, both of his hands are holding the metal object.
When Boimler is animated his sideburns are pointed in the traditional futuristic Star Trek style, but when he switches to live action his sideburns are squared in the present-day style.
When live-action Boimler first tumbles out of the portal his red uniform jacket has ridden up in back to reveal a black undershirt. In the far shot of him lying on the ground unconscious his red jacket is still bunched up in back, but the black undershirt is not there.
Number One shows Pike that Boimler's badge also functions as a communicator and he is unfamiliar with such technology. Pike encountered Starfleet officers using communicator badges ("combadges" as they are colloquially known) in the second season of Star Trek: Discovery (2017), when Section 31 operatives were seen using them.
However, that entire mission has been highly classified, Pike would be legally barred from mentioning that detail and avoids the subject by saying "But flipping it open's the best part".
When Boimler is sitting with Chapel and Ortega in the mess hall, the pointed lower front of his uniform jacket curls upward revealing a roughly-sewn seam on the inside of the costume.
When Boimler awakes in Sick Bay and recognizes Pike, he thinks he might be in a Holodeck and calls for the computer to end program. When that doesn't work, he then gentle pokes Pike to see if he's solid. However, it's been established that Holodeck characters can and usually are tangible. So poking a supposed Holodeck character would prove nothing.
When discussing the portal while still aboard the Cerritos, Boimler says it was discovered by the original crew of the Enterprise (NCC-1701), referring to Pike's crew and the historical events he would later participate in. Pike was not the original captain of the Enterprise. The original captain was Robert April, with a different command crew (Pike served as his first officer for a time before both men were promoted).
When discussing the age of exploration in which they live with Chapel and Ortegas, Boimler refers to "aliens that are still... alien" and gestures towards a pair of Andorians. Andorians were co-founders of the Federation and would not have been considered "alien" to the crew of the NCC-1701 a century later.
Captain Pike refers to the Star Trek: Lower Decks (2020) ensigns as "Mister Boimler" and "Miss Mariner." Naval tradition, which Star Trek has generally followed, dictates that officers be referred to as "mister," regardless of gender.