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.
The direct sum is, therefore, what you always do "under the hood" when you build up higher dimensional spaces from lower dimensional ones.
Similar reasoning applies to 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 $A_1\oplus A_2$).