Yes, The whole Catalina and iOS13 thing has been a mess for me. I wish I had stayed with High Sierra. Anyhow, I had the same problem and I found a solution on the iPhone:
1) Go to Settings->Messages
Look under "Send & Receive"
I saw only my personal phone number here and not my apple ID or other emails.
2) Go to Settings->Messages->Send & Receive
You can then select to login to your account using your Apple ID. Once the login took place all my stuff started working again. The list shows my phone number and my several apple accounts (the old .mac, then .me, and finally .icloud). Yours might be slightly different of course. It appears that the phone ultimately ends up creating the bridge between SMS and iMessage. I was under the impression this took place somewhere else, or perhaps this allows the bridge to happen somewhere else. Whatever the case, it worked for me.
Good luck.