At school we are taught that functions like $f(x) = x+1$ are linear. But according to the formal definition, in order to be a linear function it must exhibit the properties of additivity and homogeneity of degree 1. $f$ does not exhibit these properties. For example, for additivity:
$f(x+y) = (x+y) + 1 = x+y+1$
But $f(x)+f(y) = (x+1) + (y+1) = x + y + 2$
This means
$f(x+y) $ does not equal $ f(x) + f(y)$, further implying that $f$ is not additive. This is contrary to what I have been taught in the past, namely, that $f$ is a linear function, more specifically, that all continuous functions with degree $1$ are linear, by virtue of their graphs looking like a line. Explain the discrepancy.
In linear algebra this is not considered a linear function, since it does not respect linear functions axioms, as you noticed. This type of functions is sometimes referred to as "affine" functions. You are correct in observing that in both cases, the graph represents a straight line.
The underlying idea is that linear functions respect the structure of a vector space (or linear space) where the origin is a fixed point. Conversely, in this case, we are working with affine spaces where the origin is not fixed. In short, "affine functions" can be seen as translations of linear functions.