I think one issue is that if the parents are particularly invested in stereotypical ideas of gender, you can get issues with in-laws. We still have the ingrained idea in our society that men should be good providers, earn a decent wage, work hard, be able to drive, fix things and be "stable". Of course, most parents also want their daughter to be with someone loving and who treats them as an equal. But I still think that even just subconsciously, a fair amount of parents would like a SIL to be practical and hard working. Lots of women have children and go part time, or SAH for a time, which has a knock-on effect to their long term earning potential, and I think we still have this idea that a man should be able to provide for his family. The majority of issues I've ever heard of between FILs and SILs, whether it's the FIL stepping in and treading on toes or little criticisms etc, have been where the FIL has felt the SIL was a bit useless with money, wasn't able to put up a shelf, perhaps had MH issues or anything where they fell short of the FILs idea of manliness. However, in my limited experience, I've never seen much jealousy between FILs and SILs. Possibly in part because girls are socialised to stay close to their parents, and look up to their dads, there is no notion of having to "cut the apron strings" with girls, and also because being a father just has a different status to that of a mother in our society.
I think the DIL/MIL relationship is far more complex. Not always, of course, but we still have ingrained the idea that women should be competent in the domestic sphere, nurturing, caring, selfless etc. A fair amount of MILs will have worked part time or SAH, and/or done most of the cooking, cleaning and "wifework". Whilst their husbands had the opportunity to progress at work they had their children and the home and put everything into that. I'm making big generalisations just to make a point, I hope this makes sense, and I know "not all MILs" but I'm talking specifically about MILs who have traditional ideas about gender etc, and/or MILs who haven't pursued careers. Anyway, whilst your children are young, you have a focus. They need you. But as they get older they become more independent. If you haven't gone back to work and/or haven't got other interests, then you might find you now feel a bit redundant. If you have a daughter, you might expect her to remain close to you. Perhaps she will have children and will ask your advice, or rely on you for childcare. Perhaps you will bond over shared experiences. You don't feel threatened by her partner, because society expects women to be close to their mothers.
If you have sons, particularly if you have no daughters, it's a different situation. You've heard all the "son's a son till he takes a wife" and "must cut the apron strings" and you worry that he won't need you anymore once he has a wife. After all, she will go to her mum for advice, childcare, meeting for coffee etc if DC come along, and your DS will likely be working full time. Or they might move nearer to her parents. Your DIL might feel like the gatekeeper to her family, including your DS, and you may be really different people, who have very different ideas. You have lots of experience with raising children and running a home and you might feel you can be useful in that way.
It's only anecdotal of course, but all of the MIL problems I've encountered have been where the MIL had all boys, not a lot of interests, and had pampered her sons and husband. I know several MILs who more or less waited on their sons hand and foot then gave their DILs hell for not doing the same, or for "taking their son away". Sadly, I think they all, consciously or not, feared losing their sons in adulthood so they treated them like kings in the hope that any potential DILs wouldn't measure up. In the case of my own MIL, she resented any independence at all from DH and she resents my own domestic competence. She would prefer to have DH taking his ironing home at the age of 35, or that neither of us could boil an egg. It's sad, because I don't see her as competition in any way, but she's spent DH's whole life trying to make him dependent on her and telling him that women should look after men, and the result is that he's actually run a mile from her because her jealousy and desire to "keep him" just means she infantilises him and tried to destroy all his relationships with women.
I have sons, and I feel it's so important to help them be independent and not to stifle them. I hope that they will actively want to spend time with me as they grow up, and that any DILs will recognise that I'm an ally, who brought my sons up to be good, feminist men.