Iterating Bayes rule over time $\require{cancel}$
In a online bayesian inference procedure one is iteratively changing the prior with a new posterior, calculated given a new set of observations.
Does it mean we capture time dependence in this manner?
What actually changes, our belief about the system or the system itself?
Can you say that time dependence emerges in our model from these sequential prior updates?
EDIT Adding more rigor:
Variables: $X$ - input, $Y$ - output, $\Theta$ - hidden parameters
Observation $Y_i$ is independent of $Y_{i-1}$ given $X_i$ and $\Theta_i$ $(Y_i \bot Y_{i-1}|X_i,\Theta_i)$
Update rule for posterior is:
$P(\Theta_{i+1}) = P(\Theta_i|Y_i,X_i) = \frac{P(Y_i|X_i,\Theta_i)P(\Theta_i)}{P(Y_i|X_i)}$
So we say that even if our model is time independent, time dependence can be captured implicitly within $X$ and $\Theta$: $X_{i+1}\cancel{\bot} X_i$ and $\Theta_{i+1}\cancel{\bot} \Theta_i$.
Is it a correct assumptions to make? 
 A: 
Does it mean we capture time dependence in this manner?

No, time is not a random variable, so it is not in the prior though it can be part of the data.  Time can be part of the likelihood and if it is then it can capture any time dependence.  It can also be captured if time is an implicit function of the likelihood and the data is not exchangeable.

What actually changes, our belief about the system or the system itself?

If the system changes then the form of the likelihood needs to change unless by the system itself you mean that the state is changing within a model.  The belief is being updated by the posterior.  If part of the inference is about the state of the system, then the posterior captures this.  
An interesting example of this, where the likelihood needed to change, is the Corrupted Blood incident in the World of Warcraft.  A game update introduced a communicable disease into the game that killed whole cities with nobody able to figure out how or why it happened for quite some time.  The introduction was an accidental result of a software update.  They introduced an illness in a very narrow and specific encounter, but pets involved in the combat would get it too.  If a player happened to dismiss their pet before they died from it the pet still had it when they were resummoned.  If others were around when they came back, they got the disease and spread it.  

Can you say that time dependence emerges in our model from these sequential prior updates?

No, time dependence is only present if the data are not exchangeable.  The fact that data is revealed over time is a separate idea from time dependence.
