A dimension, as used in physics and engineering, is often used to describe a unique, well-defined attribute to which a quantitative value or function can (at least theoretically) be assigned. Specifically, as described below by the BIPM, this attribute must be able to be described/used within a system of non-contradictory equations relating the relevant system of quantities.
I apologize in advance for the wishy-washiness of this definition, but it is the best that I can come up with. I highly recommend reading section 1.3 of The SI Brochure. It might also help to read an IT perspective like this section on n-dimensional arrays in Numpy.
Dimensions in common parlance refer only to spatial dimensions (and sometimes time) but practically any attribute can be called a dimension depending on the circumstance. Forthermore, the properties of dimensions that are familiar in the spatial case can often be extended to other situations.
For example, if two dimensions are orthogonal, it means that one cannot be described, even in part, by reference to the other. A vector on the xy-plane, for example, can be described in terms of $\hat x$ and $\hat y$; but $\hat x$ and $\hat y$ cannot be described in terms of eachother. This concept is often extended in fields such as quantum mechanics where wave functions can be treated like dimensions and tested for orthogonality.
I should have included it origianally, but a good source for definitions used in science and engineering is the International Vocabulary of Metrology (VIM). According to this source:
dimension of a quantity dimension
expression of the dependence of a quantity on the base quantities of a system of quantities as a product of powers of factors corresponding to the base quantities, omitting any numerical factor
where a "system of quantities" is defined as
system of quantities
set of quantities together with a set of noncontradictory equations relating those quantities
Clearly the definition of "dimension" (as I understand your question to mean it) is quite broad and might more properly be referred to as "base quantity."
quantity in a conventionally chosen subset of a given system of quantities, where no subset quantity can be expressed in terms of the others