The last time the Greek alphabet was truly overhauled was millennia ago, when a version tuned for the Ionian dialect (known as the "Euclidean alphabet" after the archon who championed it) was propagated to most of Greece. It didn't quite have a letter for each phoneme—it didn't have separate letters for all the long and short vowels, for example, and it wasn't until a while later that people started indicating the pitch accent—but it was pretty well suited for Ionian, and pretty much okay for other dialects.
Then the language continued to evolve for thousands of years, but the alphabet has never again been overhauled to the same extent. As part of that evolution, the
/b/ sound in Ancient Greek started being pronounced as
[v]. (Somewhat later, the sequence
/mp/ came to be pronounced as
[mb], as in πέμπω "send".) The alphabet still worked fine at this point; they didn't have a
/b/ phoneme separate from
/v/, so having a single letter beta worked fine.
Except then, Greek borrowed a bunch of words from Turkish (and other places) containing a
/b/, and they needed a way to represent it. The only place the
[b] sound appeared in native Greek words was after
/m/, like in πέμπω, so they started using the sequence μπ to indicate
/b/ in loanwords, like μπρούτζος "bronze".
If the alphabet ever got thoroughly overhauled again, they might split beta into two separate letters, one for
/b/ and one for
/v/. That's what Cyrillic did: В is
/v/ and Б is
/b/. But there's a lot of cultural resistance to making a big change like that, since it breaks continuity with centuries of old texts and inscriptions connecting modern Greece to Homer and Socrates. For now, using μπ to write both
[b] doesn't cause enough of an issue to overcome that resistance.