The difference between kids and adult brands is typically not the fluoride content (which our dentist once paraphrased as „adult teeth need adult fluoride“), but the flavor profile. „Adult“ types often are very minty for fresh breath, which children often perceive as „strong“ or „sharp“. Kids toothpastes are typically milder with a fruity taste.
Additionally, some adult toothpastes contain either abrasive particles or bleaching substances, which may be too harsh for the not-yet-fully-hardened permanent teeth which have just broken through. Those types are not suitable for children.
So if you stay away from the „stain removing“ or „whitening“ toothpastes, and if your child can handle the „sharpness“, they can use the adult brands. Use just a pea-sized dollop, as often stated on the packaging.
But frankly, I’m happy to pay a bit more for the kids brand if that means my child diligently brushes (instead of rushing through the routine because the toothpaste is uncomfortable). I think in the long term, paying a few cents more for the toothpaste now can save a lot of money in the long run.
In a pinch (while traveling, for example), sharing your toothpaste is fine.