Entropy is a function of the distribution. That is, the process used to generate a byte stream is what has entropy, not the byte stream itself. If I give you the bits 1011, that could have anywhere from 0 to 4 bits of entropy; you have no way of knowing that value.
Here is the definition of Shannon entropy. Let $X$ be a random variable that takes on the values $x_1,x_2,x_3,\dots,x_n$. Then the Shannon entropy is defined as
$$H(X) = -\sum_{i=1}^{n} \operatorname{Pr}[x_i] \cdot \log_2\left(\operatorname{Pr}[x_i]\right)$$
where $\operatorname{Pr}[\cdot]$ represents probability. Note that the definition is a function of a random variable (i.e., a distribution), not a particular value!
So what is the entropy in a single flip of a coin? Let $F$ be a random variable representing such. There are two events, heads and tails, each with probability $0.5$. So, the Shannon entropy of $F$ is:
$$H(F) = -(0.5\cdot\log_2 0.5 + 0.5\cdot\log_2 0.5) = -(-0.5 + -0.5) = 1.$$
Thus, $F$ has exactly one bit of entropy, what we expected.
So, to find how much entropy is present in a byte stream, you need to know how the byte stream is generated and the entropy of any inputs (in the case of PRNGs). Recall that a deterministic algorithm cannot add entropy to an input, only take it away, so the entropy of all inputs to a deterministic algorithm is the maximum entropy possible in the output.
If you're using a hardware RNG, then you need to know the probabilities associated with the data it gives you, else you cannot formally find the Shannon entropy (though you could give it a lower bound if you know the probabilities of some, but not all, events).
But note that in any case, you are dependent on the knowledge of the distribution associated with the byte stream. You can do statistical tests, like you mention, to verify that the output "looks random" (from a certain perspective). But you'll never be able to say any more than "it looks pretty uniformly distributed to me!". You'll never be able to look at a bitstream without knowing the distribution and say "there are X bits of entropy here."
