You have identified a existing premise of morphological theory, so yes, one can say that words which share a morpheme also have shared semantic properties. That premise is not necessarily accepted by all theorists, but it is very widely accepted. It follows from the theory of compositionality, that morphemes have some form *(e.g. -k, -veg- etc.), and a syntactic and semantic function, signalling tense, case, number, gender agreement etc. or some lexical property.
The position that each morpheme makes a semantic contribution (including indirect semantic contributions in the form of syntactic signalling which itself contributes a bit to semantic interpretation – such as case marking) is a pretty weak claim, in light of the fact that we see many cases where the full meaning and use of a word cannot be computed from the purported meaning properties of the constituents. Semitic root derivation as you mention is an example. The pertinent question that theorists ask is, to what extent can decomposition into morphemes be justified when the meaning of a combination is not regularly computable from the meaning of the sub-parts? That has been a long-term unresolved squabble in linguistics.
English has a number of verbs (ultimately taken from Latin) that are treated as being bi-morphemic, for example submit, remit, transmit, permit, emit, admit, commit; subsume, resume, assume, consume, presume. The analogous word *transume is lexicographically treated as "archaic" (meaning, "there used to be such a word"), and "exume" is a spelling variant of exhume which involves a different supposed root. The Latinate prefix-plus-root constructions are sufficiently unpredictable in terms of possible combinations and semantic interpretation that many people treat subsume, resume, submit, remit as monomorphemic roots. The extent of required predictability is an fundamental variable in theories of language. Under the rubric of "Lexical phonology and morphology", many linguists found a way to have their cake and eat it too, so that the desiderata of transparent shared meaning and productivity were subordinated to the desiderata of maximally identifying "patterns", and thus it became acceptable again to say that per-mit is bi-morphemic, even though the meaning of "permit" cannot be computed from the meaning of "per" and the meaning of "mit" – the meaning itself must be stored in the lexicon. That theory encompasses both bimorphemic words whose meaning is semantically compositional, and bimorphemic words whose meaning must be stored. In that theory, there is not necessarily a computed semantic relationship between words that have a stared morpheme.