That doesn't mean much, Harry Kim has been in charge of the bridge many times but Janeway still kept him an ensign for the whole seven years. In fact, I wouldn't be surprised if she had kept him that way for the twenty-six years it took them to get back in the other timeline and was only promoted AFTER their return.
Ensign Kim was only in charge of the bridge during night shifts, when the regular command crew were asleep, and that was only because he volunteered. Nilsson seems to be in charge whenever Saru and the XO-du-jour are awake but just off the bridge.
Back in the TNG days, Dr. Crusher would also volunteer for night shifts (see the episode "Thine Own Self"), although she had already taken what was then called the "Bridge Officer's test" to become a full Commander.
It's possible that Nilsson isn't technically qualified to serve as XO, in that she's an engineer who was only given the Conn because of her rank and seniority amongst the bridge crew. Tilly might be the only one, besides Saru and Burnham, who was enrolled in the Command Training Program and could theoretically take Command officially.
That's possible. Though I should point out that in TNG, Data was the ship's Second Officer (see the TNG episode "The Defector", where Data does a "second officer's log"), despite being in the Operations division like Nilsson is. Though he was a Lieutenant Commander vs. Nilsson, who was just a Lieutenant. In TOS, I think Scotty was second officer, since he was always sitting in the Captain's chair while Kirk and Spock were off having adventures.
Of course, I imagine the real out-of-show reason is that the actress who plays Tilly is in the opening credits, and it wouldn't make much sense to give a background actor such a dominant role.

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