It is difficult to describe the amount of misinformation here.
There are HUGE differences between types, brands, and ingredients in laundry detergents. And no, mixing them is NOT a good idea, but probably won’t hurt anything.
There is a huge difference between an alkali (petroleum)based detergent powder and a natural based liquid detergents. Powders are better on grease, oil, and chemical based stains…liquids are better on organic and protein based stains like grass or blood.
But presuming you are speaking strictly about liquids, there are still big differences.
1.) All detergents need surfactants. Surfactants allow the dirt to attach itself to the water molecule to allow the dirt particles to float away with the rinse water instead of staying attached to the clothing. Some brands use anionic surfactants and some use ionic surfactants. mixing them is counter productive.
2.)Many liquid detergent ingredients do not mix well. One of the biggest drawbacks of liquids is the inability to include boosting agents and their short shelf life. This is the single reason that Pods are such a breakthru in liquid tech is because they can keep the various chemicals separated until use. This allows for the inclusion of several important water softening agents which until now have not existed anywhere but powders, and also greatly extends the usable shelf life of liquids.
3.) The single most critical difference to you as a user, is enzymes.
Enzymes are what clean clothes now that phosphates have been banned.
Enzymes are what clean, and enzymes are what cost. Cheaper detergents have less enzymes than expensive ones.
And all enzymes are not particularly or necessarily compatible with other enzymes. They can feed off and/or destroy each other.
Corporations like Dial and Proctor and Gamble spend millions of dollars in research and testing to determine the science behind dirt, soil, enzymes, fabrics, water, and machine compatibility (IE:So their products aren’t corrosive to metals or plastics and rubber). Other brands ride their coattails and copy.
Summary:
Will you notice any huge difference doing this now and then? No.
Is there any difference in detergents? Yes. Huge.