Let $A:=\mathrm{C}^*(G,R)$ be the (defined) universal $\mathrm{C}^*$-algebra generated by generators $G$ and relations $R$. As far as I can discern, there are two ways of defining such objects --- either using category theory or non-commutative polynomials and ideals.
The big ticket is that given any $\mathrm{C}^*$ algebra $A$ with $|G|$ generators that satisfy $R$ there is a unique surjective $*$-homomorphism $\mathrm{C}^*(G,R)\rightarrow A$. This is the universal property.
I have an informal way of thinking about universal $\mathrm{C}^*$-algebras (that will shortly come into conflict with my second question).
My informal intuition for universal $\mathrm{C}^*$-algebras is that we more or less form a large direct sum of all $\mathrm{C}^*$-algebras that have $|G|$ generators which satisfy $R$ and that is where the universal property comes from (or sums of) projections $\bigoplus_\alpha A_\alpha\rightarrow A_\beta$.
Question 1: How bad is this intuition? Is there something big that it misses (forget if it is formally incorrect)?
That is more or less the question on definition. Now for my doubt. We can consider a particular universal $\mathrm{C}^*$-algebra $B$ which is generated by a magic unitary $u=(u_{ij})_{i,j=1}^4$. That $u$ is a magic unitary is to say that each entry is a projection $u_{ij}=u_{ij}^2=u_{ij}^*$ and that each row and column is a partition of unity:
$$\sum_{k=1}^4u_{ik}=1=\sum_{k}u_{kj}.$$
In the compact quantum group community (say p.8) this universal $\mathrm{C}^*$-algebra is the setting for the algebra of continuous functions on the compact quantum group of permutations. The definition (of compact quantum group) requires a $*$-homomorphism $\Delta :B\rightarrow B\underset{\min}{\otimes} B$ and how this is constructed in this example is to define
$$\Delta(u_{ij})=\sum_{k=1}^4u_{ik}\otimes u_{kj},$$
and the way it is shown that this is a $*$-homomorphism is to show that the matrix:
$$\left[\Delta(u_{ij})\right]_{i,j=1}^4\in M_4(B\otimes B)$$
is a magic unitary, and so $\Delta:B\rightarrow B\otimes B$ is a *-homomorphism. Then the confusion comes in because in my informal intuition from a above $B$ is the largest object with magic unitaries... but here we have $B\otimes B$ which in my understanding is LARGER again than $B$ is also a sub-object of it...
Question 2: Is there a way to resolve this confusion with my informal intuition of what a universal $\mathrm{C}^*$-algebra (perhaps by arguing that the $\mathrm{C}^*$-algebra generated by $\left[\Delta(u_{ij})\right]_{i,j=1}^4$ while living in a large space $B\otimes B$, is in fact smaller than $B$... or is my informal intuition messing up my understanding here?
Question 1: Your intuition is not only correct, but it also underlies the proof of the existence of the universal algebra.
Question 2: The universal property is only relevant in the domain. All that is asked about the co-domain is that the relations be satisfied.