What you are modelling when you use a tensor product is a space that can accommodate tuples of basis states of different kinds.
For example, say you want to describe a three-level system. You can use a three-dimensional Hilbert space $\mathcal H$ to accommodate the possible states of this system.
But what if you have two three-level systems, each one of which can be in one of its three available states? In this case you have nine possible basis states ($00$, $01$, $02$, $10$, and $11$, $12$, $20$, $21$, and $22$), and thus need a nine-dimensional Hilbert space. As it happens, $\mathcal H\otimes\mathcal H$ has just the right dimensions (you could also use any other nine-dimensional Hilbert space, only it would make the notation more awkward when dealing with local operations).
On the other hand, $\mathcal H\oplus\mathcal H$ is a six-dimensional space, with basis the union of the bases of the two copies of $\mathcal H$ (see also this post over at math.SE). You could write this basis as
denoting with $|i'\rangle$ the $i$-th basis element of the second copy of $\mathcal H$.
Clearly, this does not describe a system that is obtained by combining multiple elementary systems. Rather, it can be used to describe how a high-dimensional system is "composed" of smaller-dimensional ones.
Indeed, one can always think of an $n$-dimensional Hilbert space as the direct sum of $n$ copies of one-dimensional spaces.
In other words, in some sense, the direct sum is what you always use to build up high-dimensional spaces.
Similar reasoning goes for Hamiltonians or other operators. As an example, if you have a five-dimensional space $\mathcal H$, and two operators $A_1$ and $A_2$ operating on three- and two-dimensional systems, respectively, then $A_1\oplus A_2$ is a valid operator acting on states in $\mathcal H$. This represents an operation which does not correlate the first three and the last two modes (because of the block structure of $A_1\oplus A_2$).