The main difference between liquid soap and dish washer detergent is price. So not a good idea in general, but only because liquid soap is more expensive.
One recipe for liquid soap is given here – note olive oil, coconut oil, essential oils. Some of the ingredients for a well-known brand of washing liquid are listed here – note Sodium laureth sulphate, Alcohol denat, sodium chloride.
For the first of these three Wikipedia has “SLES is an inexpensive and very effective foaming agent.” For the second, consumers are used to alcohol (ethanol) being expensive, but that is only because of the tax. Denatured (methylated spirits) it is very much less expensive. Diggers 1L is offered for AUD 3.30. The third is common salt.
Consider instead the first of the natural liquid soap ingredients. Aldi is offering one litre of olive oil for AUD 5.99. Coconut oil is much more expensive still - AUD 20 per litre at Woolworths.
However, while Cocamidopropyl betaine features in detergents it is derived from coconut oil.
According to brown thumb mama “soap is natural and detergent isn’t”. That was the case around the time detergents were being developed (say 1930-50) but now most ‘soaps’ are synthetic products.
Another significant difference is the ecological effects – see Wikipedia. Soap made from natural products is far friendlier to the environment.
However, in terms of cleaning power they both work the same way. They are surface active molecules that cling to dirt at one end and to water at the other, and then get flushed away (with the dirt) on rinsing. Though there are some differences in how the two respond in ‘hard’ water (that is, where the mineral content is high).
In terms of bactericides, some dish washing liquids are advertised for their disinfectant properties but, as with much else in the world of soaps and detergents, there is a LOT of hype. “Soaps” (some TV programmes) are not called what they are because the industry giants could not afford millions to promote products earning them billions. Dirty dishes are ones we have just eaten off – they are not an environment that supports germs the way the human body does. Do we wash our hands in washing up liquid after a trip to the lavatory? No we use ‘soap’ (at least, I hope you do) because that kills germs.
So, any form of detergent is good enough for dishes and liquid soap more expensive - but much more environmentally responsible.
From a travel aspect however washing up liquid is generally more concentrated than 'liquid soap' so might be more practical if to be carried around. The greater concentration is also a positive for washing up liquids to offset some of their environmental negatives.