What you are asking about is called bioavailability, which is the term for how much of a consumed substance is actually taken up by the body, and bioequivalence, which is about whether two products are used the same way in the body.
For protein, bioavailability is also called Biological Value (BV). The BV of an egg, for example is somewhere around 93 (see column 5). For cooked chicken, a bit over 70.
It's hard to find sources for protein powder/shakes that don't seem biased (being from the manufactors of the protein, or bodybuilding sites), but here's a study from the Journal of Sports Science and Medicine: Protein - Which is best?. It uses an adjusted scale, where a whole egg has a biological value of 100. In that scale, since whey protein has a higher BV than whole egg, it comes out as 104. Soy protein and casein have BVs in the mid-70s, a bit lower than beef.
As for whether they have the same effect, well, what's different betrween different sources of protein is composition. Proteins are made from building blocks called amino acids, of which there are 20 in food. Many of those can be converted from one to the other or made from other things we eat, but 9 amino acids are called 'essential' and have to be eaten. Different foods, including protein powders differ in what amino acids they have. Usually, protein powders will contain these. How much they contain will depend on the product.
Keep in mind that no source of is 'perfect' in that regard, either. A bean does not have the same amino acid composition in its proteins as a human body does. Neither does whey protein. Deficiency in essential amino acids isn't pleasant, but it is rare in people who aren't generally malnutritient and consume enough protein overall, from more than one source.
Basically, a lot of the same applies for vitamins. They aren't perfectly bioavailable when digested from food or from a multivitamin, sometimes one is better, sometimes the other. Answering this for all vitamins is way too broad, and it probably also depends on the product. The paper Multivitamin and multimineral dietary supplements: definitions, characterization, bioavailability, and drug interactions goes into a few of the problems and general concepts.
For example, it looks like the bioavailibility of folic acid taken up from dietery supplements is higher than that from food, and especially outperforms spinach and yeast (warning: low number of subjects in study...). For vitamin C, Synthetic or Food-Derived Vitamin C—Are They Equally Bioavailable? summarizes:
In contrast, all steady state comparative bioavailability studies in humans have shown no differences between synthetic and natural vitamin C, regardless of the subject population, study design or intervention used. Some pharmacokinetic studies in humans have shown transient and small comparative differences between synthetic and natural vitamin C, although these differences are likely to have minimal physiological impact
And goes on to say:
Although synthetic and food-derived vitamin C appear to be equally bioavailable in humans, ingesting vitamin C as part of a whole food is considered preferable because of the concomitant consumption of numerous other macro- and micronutrients and phytochemicals, which will confer additional health benefits.
So whole foods are preferable, but the vitamin C that you get from the supplements itself isn't any better or worse than the vitamin C an apple contains, and certainly isn't unhealthy.
For some people, getting all vitamins they need from their diet is either impossible or very hard (vitamin D and iron deficiencies are rather common) and that's where supplements like the protein bars you use come in. As pointed out by Atl LED in the comments, though, for vitamin supplements to work it's important to take care of how the multivitamins need to be taken. If they need to be taken with food for maximum absorption, that's important or else you won't get the full benefits.
If protein bars are judged as 'unhealthy' it should be because of things like their sugar content, but it's certainly not true that the proteins or vitamins in them are unhealthy or that we can't use them at all.