My R5M2 batteries have a white stripe on the front, otherwise you are right, they look very similar. I've taken all of the older batteries out of my case and backpack just for this reason.
Be sure, they are similar in their physics. Just canon has wrote a different Device Id in the balancer/management that the Body can readout the vital data and type. I assume that Canon uses cells with a litte bit higher enery density. I had seen that Sanyo produces this since a two or three years. The capacity of the NH and P cells are the same. The difference between then NH and P battery is, that the P supports a discharge rate of 6A. If the Body consumes that 6A for around 10 minutes, the battery is discharged. I assue that the modes that are not available with an NH battery in the Body, will requires more power, that the older batts not distributes.
Well, i don't see a generally mutch higher price for the LP-E6P as the LP-E6NH. If you are able to pay for a new R5II, the batteries should't be a further hint.
In all of my time of photographer, i had never bought a compatible battery for my Bodies. Some of the batteries are going lower after around 7 years of usage. The magic of the relative good condition for their age is, that i don't discharge them to the shutdown point of the Body and i recharche them up to 40% if i don't need them longer. I have two batteries at 80% level in the bodies.
and yes, this management style requires to plan extended photo excurses previously. But the batteries lives longer if the not charged up to 100% and not discharged lower that 20%. All of them is not new or a magic. But this saves money in an easy way. None of the batteries was ever cheap and sometimes i look jealous to patona or other brands with quite lower prises. Not that i don't trust them, but my bodies was not cheap and i like them, even some of them are very good with mirrors. If one time, the first battery will be death, i replace them by a LP-E6NH again. That are good enought for the R5, R, 5DM4, 5DS, 80D. They all use the same battery type.