If the i.i.d (independent and identically distributed) assumption holds, shouldn't the training and validation trends be exactly the same?
No, not necessarily. Let me explain why.
If you assume your samples (aka examples, observations, data points, etc.) are i.i.d., this means
that they come from the same distribution, e.g. a Gaussian $\mathcal{N}(0, 1)$ (the identically distributed part), and
that they are independently drawn from it, i.e., intuitively, each sample provides the same kind of information independently of the others
However, even if samples are independently drawn from a certain distribution, they can be different. For example, if you draw a sample $x$ from $\mathcal{N}(0, 1)$, an operation often denoted as $x \sim \mathcal{N}(0, 1)$, $x$ could have the value $0$, $1$, $13$ or $50$ (or any other number), so they could be variable, although your samples will tend to be mainly around $0$, because that's where your Gaussian puts more density (and your standard deviation is just $1$). If your standard deviation was higher, then there would be even more variability in the sampling process.
So, if you assume that your samples are independently drawn from a certain distribution, it doesn't mean that you will always get the same pattern of samples. In other words, you can still have variability in your samples, and this also depends on the distribution you sample from.
To answer your question more directly, there's a chance that your training data and your validation data don't necessarily have the same patterns, even if the independence assumption holds. Therefore, the training and validation trends (and I assume you mean e.g. the performance) are not necessarily the same, but, although this may also depend on the training method, I would say that they shouldn't be very different (if the assumption holds) because, intuitively, each sample should be as informative as any other sample (independence assumption).
Do we work with the fact that there is a certain degree of wrongness to the assumption or am I interpreting it wrongly?
It is often convenient to make the i.i.d. assumption even, if it doesn't hold, for several reasons:
your training procedure may converge faster (because, intuitively, each sample will be as informative as any other sample)
your models may be simpler (e.g., in naive Bayes, you make the i.i.d. assumption only to simplify the model and, in general, the mathematical formulations)
Sometimes, if it doesn't hold, your training procedure can be highly affected. In those cases, you can find workarounds and try to make it hold. For example, the usage of the experience replay in deep Q-learning is an example of a trick used to overcome the dependence of successive samples, which causes learning to be highly variable. See this question Why exactly do neural networks require i.i.d. data?.
The answers to the question On the importance of the i.i.d. assumption in statistical learning on CrossValidated provide more information and details, so you may want to have a look at it too. Here's another answer, which is related to shuffling and how it can or not make the independence assumption hold, that I highly recommend that you read.
