User rhettf, writing at the Apple Support Community:

Since updating to Mojave, iMessage on Mac is no longer able to send or receive SMS messages. You get the red exclamation point. I tried signing out of iMessage on the Mac. And also toggling the Text Forwarding to device on the iPhone. Is anyone else experiencing this problem?

My experience is a bit more nuanced than this on Mojave. When I send a text message to an Android user from my Mac on macOS 10.14, the message goes through — the recipient receives it successfully. But the Mac doesn’t recognize this as occurring. It eventually shows an error in the form of a red exclamation mark, as rhettf describes. Moreover, when the recipient replies, I get the reply on my iPhone but I do not get the reply on my Mac.

In other words, on Mojave, sending an SMS technically works, but receiving does not, and neither of these things work from a UI standpoint.

Update: I found a fix that works for my computer. As you can tell in the link, a lot of people have this bug still so my solution isn’t the canonical answer, but it did fix it for me. SMS messages now forward correctly to my Mac on Mojave. The secret was signing out of Messages on the Mac (Messages -> Preferences -> iMessage -> Sign Out) and on my iPhone (Settings -> Messages -> Send & Receive -> Apple ID: {email} -> Sign Out). After that, I signed back in, taking care to use the same email address everywhere. Technically I have a total of 3 different email addresses associated with iCloud, all with the same account, but you have to log in using the primary email address everywhere, and then it works. If you’re not sure which is your primary email address, you can view it here. The reason that it appeared that this problem began occurring on Mojave is because somehow my email address for Messages got changed to a different one. Maybe I messed that up when upgrading to an iPhone XS. In fact in hindsight, I’m not sure if the problem was upgrading to Mojave or getting a new iPhone. I did them both just a few days apart.