I can only come up with Sears. My parents always had Kenmore appliances and we kids were always dressed in Sears clothing. So when I got married and bought a house, most of our appliances were Kenmore - vacuum too!
But that has changed as of late. Our Sears store looks like a ghost town.
Sure they are. IIRC it was not a chain in the usual sense but individually-owned motels that banded together to they could share reservation services and the like.
I can’t think of anything my parents used that I do. Mom always made her biscuits and pancakes from Bisquik and I make mine from scratch. She was jealous and it’s the one thing I was better at, cooking-wise. I just said if you’ve got the components handy it doesn’t take much longer than opening a Bisquik box, especially if you have a pastry cutter for the shortening on the biscuits.
Geez, almost everything I can think of was a poor brand, gone, or just not for me.
But: there’s Dial soap, the old “gold” bar.
State Farm insurance
We were a Crest family, my wife’s was a Colgate family. Forty years after each of us left home, we both still hold by our family’s preferences.
My uncle worked at the Ford plant for over 50 years. My dad only drove Fords. Suffice it to say, I drive a Ford.
Kramarczucks for all holiday meats. I tried a different butcher once. Once.
Lysol for cleaning the bathroom
Land o’ Lakes butter
Kemps for pretty much everything else dairy
Dawn dish detergent, Cascade for the dishwasher
Listerine mouthwash
Canada Dry ginger ale
Duke’s Mayonnaise, and Duke’s only. All the rest are dead to me.
White Lily or Martha White flour, but you hardly see Martha White in the stores anymore, so I go right for the White Lily because, what? You expect me to make a decent biscuit with some Yankee flour like Pillsbury? You better get on away from here!
And Dixie Crystals sugar, I don’t care if it costs more…
Showin’ a theme here, ain’t I? :o
Well you did ask about family influence, so…
And no, you won’t find a scrap of Northern toilet paper in my house, either! So there, that’ll show ya! 
Oh, and LeSueur Peas in a can, always to be served with mashed potatoes, bird nest style.
Slight hijack: you know why I was reminded of this childhood staple of my mother’s pantry? Because right as soon as submitted the above Southern-centric post, I swear on the grave of Margaret Mitchell that the ad banner at the bottom of my screen said “Southerners love our small sweet peas” with a LeSueur can…there’s a ghost in my machine for reals!
The only ones that come to mind are going to be completely esoteric for 99.99% of everybody. Here, we’re partial to Alexandra’s pierogi (as opposed to Kasia’s, which will do in a pinch, and Mrs. T’s, which won’t.)
Oh, and not-as-esoteric, I’ve picked up on Old Style as being my “lawnmower” beer of choice from my dad. First beer I ever tasted. I remember being four or five and him and my uncle watching boxing matches, yelling at the TV, and drinking Old Styles. I would not stop pestering him until he let me have a sip.
But that’s about all that I could think of. Oh, and maybe Kosciuszko mustard as my brown mustard of choice.
Originally Laura Scudder’s peanut butter, although both my parents and I eventually switched to Trader Joe’s store brand. I’m still loyal to the style though. If it doesn’t need to be refrigerated, it’s not peanut butter. If you don’t need to spend half a minute stirring in the quarter inch of oil sitting on top before spreading it on bread, it’s not peanut butter. I won’t be passing it on to my kids as they think “Dad’s peanut butter” is gross.
When I did an estate sale I was surprised to see how loyal my parents were to Kenmore. Every appliance in the house (washer, dryer, refrigerator, oven, microwave) was Kenmore. I’m more of a Maytag kind of guy.
SOS pads. No Brillo or store brand in our family.
I usually buy Campbell’s soup - if I even want canned soup. Wait, my parents had no brand loyalty on canned soup either - it was often the store brand. OK then, my real answer is “nothing”. The closest thing my parents had to brand loyalty was to ignore the brand and buy whatever was carried by someone local who they liked. I guess I’ve kept that “buy from who you like and trust” at least a little bit, I don’t know. It’s different in a large city though.
Duke’s mayonnaise, Peter Pan Peanut Butter, Premium Saltines, Kiwi Shoe Polish, Gillette razors and blades. The peanut butter was the cause of our first marital argument. Oh, Mahatma rice as well.
I had to finally abandon Mama’s Mahatma rice in favor of Uncle Ben’s. Hated to give up the tradition, but I just don’t like the way it pops open on the ends is too sticky. In the end I had to accept that sometimes there really are better choices.
I realized after posting that I have no idea how my inherited choices compare to others. I don’t know if they cost more or less or if others may be better. The exception to this is Duke’s. All other mayonnaise is trash.