It's very common that there are multiple adults in a child's life who parent differently. These can be explicit differences in rules (I allow jumping on the furniture, you don't), or more subtle differences in approach (I'm fine with kids trying to persuade me something should be ok, not everyone is). Adults often get in conflict over these, and I think this is generally not worth it.
There are two main reasons I've seen for parents wanting others to take a specific approach with their kids:
You want consistency. I'm strongly in favor of predictable parenting, where kids know what the boundaries are and can predict your reactions. Kids are smart, however, and are very good at learning that different adults/places/situations have different systems.
You want a specific approach. There are a lot of ways people parent, and they mostly work fine, especially in moderation. Perhaps the aunt lets the kids have lots of sweets, grandma requires shoes outside, or the babysitter won't take them out if it's raining. While these aren't the decisions I would make, it doesn't matter that much. Now, I wouldn't want anyone who looks after my kids to enforce discipline by hitting them, and neither would I be ok with someone letting them play unattended by deep water: some things do matter. But the conflicts I tend to see are over much smaller differences.
One place where this can be more of a problem is when multiple adults with different styles are present at the same time. I think the main thing you want is consistency: if the parent is present you go by their rules, or perhaps you go by the rules of whichever space you're currently in. This does mean that if adults are often ambiguously in charge of the same kids it's worth putting effort into harmonization: different parents having different rules doesn't work very well. I expect this to come up if/when our housemates have kids.
There's also the issue of cross jurisdictional precedent ("grandpa lets me do it!") The most common way to handle this is just "grandpa has different rules", and leave it there. This is fine, but in my particular parenting style I'm happy to get into the details if the kids want to. Why is grandpa ok with it? ("That much sugar all the time wouldn't be good for you, but you're with grandpa rarely enough that I'm not worried.") Sometimes I don't know why different adults do things differently, in which case I'll say so, and suggest they bring it up with the other adult if they want to learn more.
(Story time: when the kids go off with their grandpa, it is a "Ricci adventure" and it's his rules that apply and not ours. This summer we were exploring Wareham (MA) together and the kids wanted to get ice cream. Unfortunately, they had already had their whole before dinner sweet, with some kind of sweet breakfast. They were pretty sad about this. After some thinking, Lily decided to ask Ricci whether this counted as a "Ricci adventure", and he said yes, after which she asked if this meant they could have ice cream. Rick confirmed with me that I was fine with it being his rules, and the kids were happy to have ice cream.)