From an interlingual perspective, it's anomalous that English has two different words for this. German has only "Basis" and French has only "base". These are applied without difference to vector spaces and topologies.
Math history isn't my strong point, but my impression is that topology was more strongly influenced by French mathematicians (Poincaré, Analysis Situs) and the theory of vector spaces was more strongly influenced by German mathematicians (Hilbert space, Hilbert basis). It may be just a coincidence, but it seems at least possible that the two slightly different words already existing in the English language were preferentially used to coincide with the language in which people were predominantly reading and writing about these things at the time.