The modern use, as I understand it, is as an abstract noun referring to an abstract non-physical entity with certain idealised properties. This contrasts with the medieval use we find, e.g., in Aquinas here which retains the sense of the adjective, namely as signifying that (physical thing) which has continuity, rather than the abstract feature of contuinity itself (whatever that means). 'Continuum' in Latin, like 'vacuum' is an adjective in the neuter which (in that usage) has a noun-like sense, meaning 'the continuous', or 'that which is continuous' or 'that which is unbroken'. E.g. when he says that it is impossible that "aliquod continuum componi ex indivisibilibus" he is not saying that it is impossible for some abstract object called 'the continuum' to be composed of indivisibles. Rather, he is saying that it is impossible for any real object possessing the property of continuity or unbrokenness to be composed of indivisibles.
Dicit ergo primo quod si definitiones prius positae continui, et eius quod tangitur, et eius quod est consequenter, sunt convenientes (scilicet quod continua sint, quorum ultima sunt unum: contacta, quorum ultima sunt simul: consequenter autem sint, quorum nihil est medium sui generis), ex his sequitur quod impossibile sit aliquod continuum componi ex indivisibilibus, ut lineam ex punctis; si tamen linea dicatur aliquid continuum, et punctum aliquid indivisibile.Now immediately, hearing this, there will be those who cry that Aristotle was thinking too hard about the 'real world' or the 'physical world' or something like that. As opposed to the 'mathematical world' or some abstract world of abstract things. To which I confess: I don't understand. If there is a mathematical world, in what sense is it not real? As for abstraction, I commented earlier (somewhere) that abstraction is considering normal, real things without considering the features which we are abstracting from. For example, while there is no such thing as a frictionless surface, I can still consider surfaces in respect of their shape and form, without considering properties such as friction. That is all that abstraction is. Or I can consider a triangle without considering whether it has (A) all three sides equal, or (B) two sides only, or (C) none. Now any triangle I consider must be one of (A), (B) or (C). Yet I can consider any one of them without considering whether it is such, i.e. in abstraction from whether it is any one of those three types. That is all 'abstraction' means. It doesn't mean there are any such things as 'abstract objects', as though, absurdly and impossibly, there could be a frictionless surface, or a triangle which does not have three sides equal, nor two side, or none.