"are dated January of 2023"

They would be Form Year 2023, and the tax year would be part of the code(s), such as "P" for prior, if the broker knows or cares. That's because a 1099-R is for money Out. They don't always know what happened after this step. You also want Form 5498 for money In, as due diligence for what happened, and those will have the applicable tax year on them. Also, get the statements so you can confirm these were timely.

"made a back-door recharacterization"

"Back door" relates to conversion. "Recharacterization" isn't back door. It's one of the options for corrective action. Or, are you trying to describe both Recharacterized as nondeductible Traditional IRA, and then Converted (moved into the Roth as potentially taxable)? So, that's a bit confusing.

"but not until March of 2023 for both 2022 and 2023"

You didn't explicitly state it, but 2023 also is disallowed (doesn't qualify due to income limits)?

"I have three 1099-R's."

For the year(s) money was considered Out.

"One is for the 2022 recharacterization, one is for the 2023 recharacterization"

Do they have the same distribution code?

"and one is with a distribution code of 2 for the total of both (less a small loss on the funds)."

Net loss, or by year? Having a net loss at conversion, for two years, isn't the same as determining gain/loss at each year end.There are rules for how to handle if there is loss, and there are rules for how to handle if there is gain. Unless the recharacterization(s) is the exact same amount for each year, and there were no other funds and no earnings in the/any Trad IRA, SIMPLE IRA, and SEP IRA (they get aggregated), there is no taxable gain or penalty to consider. In other words, taking the corrective action can avoid penalty, but you still need to consider earnings.

Is this the chain events:

1099-R #1: 2022 not qualified to make contributions, left them in over year end, did a corrective action before 2023 filing due date (moved the money to a nondeductible Trad IRA contribution), so the 2022 tax return might be fine (the Roth IRA mistake would be moot). You need to review it.

1099-R #2: 2023 not qualified to make contributions, recharacterized in the year they were made, so the mistake also is moot. Gain needs to be considered.

Both tax filings should include Form 8606 as basis matters (nondeductible).

1099-R #3: Conversion from Trad IRA basis to Roth. Gain still needs to be considered. A net loss on specific amounts and years doesn't define back door tax free. Only for the exact same basis with no other funds in existence in similar accounts, will be tax free; any other commingled amounts means the entire conversion pro rata conversion.

This is why the 2022 tax form might be fine (you have to examine it):

"How late can you recharacterize a Roth contribution?