Your data is completely nested. There are two Continents (A/E), each containing four Regions (N/E/S/W). It's crucial to note that the North in A is NOT the same as the North in E. In addition, within each region, there are two animal species. Again, the effects of species are NOT the same cross regions. This is a three-level nested model, which is very complicated in general.
Fortunately, your model has no random effects. Based on your descriptions, you may simplify the model by considering 16 distinct effects of species (Continents $*$ Regions $*$ Animals = $2*4*2=16$), and attribute all randomness to the error term. Therefore, you carry out one-way ANOVA twice for Weight and Size, separately. Let $Y_{ij}$ denote the Weight of the $j$th unit of the $i$th "species". Formally, we write
$$Y_{ij}=\mu_i+e_{ij},\qquad e_{ij}\overset{iid}{\sim}N(0,\sigma^2_i),\qquad \begin{cases}i=1,...,16\\j=1,...,n_{i}\end{cases}.$$ About $\mu_i$'s, see the end of this answer. You may test for homogeneity first and replace all $\sigma_i^2$ by $\sigma^2$ if the data supports equality of variances. Then the questions of interest can be translated to hypotheses.
- "American animals 1 and 2 are different from their European equivalents": $$H_0\colon \mu_1+\mu_3+\mu_5+\mu_7=\mu_9+\mu_{11}+\mu_{13}+\mu_{15}\quad \text{versus}\quad H_1\colon \ne\\
H_0\colon \mu_2+\mu_4+\mu_6+\mu_8= \mu_{10}+\mu_{12}+\mu_{14}+\mu_{16}\quad \text{versus}\quad H_1\colon \ne$$
- "within America (or EU), the animals are different depending on the regions (North, East, South and West)":
$$H_0\colon \mu_1+\mu_2=\mu_3+\mu_4=\mu_5+\mu_6=\mu_7+\mu_8 \quad \text{versus}\quad H_1\colon \ \text{not all equal}\\
H_0\colon \mu_9+\mu_{10}=\mu_{11}+\mu_{12}=\mu_{13}+ \mu_{14}=\mu_{15}+\mu_{16}\quad \text{versus}\quad H_1\colon \ \text{not all equal}$$
In total, the hypotheses impose 8 restrictions (8 equal signs) on the parameters, and I have checked that they are linearly independent (not conflicting with each other). So you may carry out an F test for them simultaneously.
If not all effects are observed, then it's possible some hypotheses are not testable because of a lack of data.
Effects |
Continents |
Regions |
Animals |
$\mu_1$ |
America |
North |
A1 |
$\mu_2$ |
America |
North |
A2 |
$\mu_3$ |
America |
East |
A1 |
$\mu_4$ |
America |
East |
A2 |
$\mu_5$ |
America |
South |
A1 |
$\mu_6$ |
America |
South |
A2 |
$\mu_7$ |
America |
West |
A1 |
$\mu_8$ |
America |
West |
A2 |
$\mu_9$ |
EU |
North |
A1 |
$\mu_{10}$ |
EU |
North |
A2 |
$\mu_{11}$ |
EU |
East |
A1 |
$\mu_{12}$ |
EU |
East |
A2 |
$\mu_{13}$ |
EU |
South |
A1 |
$\mu_{14}$ |
EU |
South |
A2 |
$\mu_{15}$ |
EU |
West |
A1 |
$\mu_{16}$ |
EU |
West |
A2 |