There is a difference in brands. I used to work at one of the largest oil companies in the world and talked to the lube engineers frequently. They pointed out that I could see cases and drums of various brands of lube coming out of our blending facility - including Ford as we had the contract with Ford at the time. What they explained is that the big oil companies make a base lube that meets the minimum spec's, and then blend in the additives required by the various companies to meet their spec's. IOW, the base for a given lube, such as Type F, is essentially the same. But, Motorcraft Type F will have certain additives, O'Reilly's Type F will have different additives or volumes of the additives, etc.

Having said that, there are specialty lubes, such as Royal Purple and maybe Trick Shift, that may be made by small companies and might be quite different from others. But, if you aren't wanting/needing to go that way then stay with the standard stuff. And, within that if you needing to save some money the house brands of many of the auto parts stores, like O'Reilly's, have their own brand that meet the necessary spec's. That's what I've used in several of the trucks I've worked on and have had no problems.