HTML signatures do work, but you have to paste a chunk of HTML-formatted signature text. For anyone looking how to get an HTML signature on Outlook app on iOS, here's how. I tried it on iOS 10.3.2 and Outlook 2.25.
The old "shake-to-undo" trick that works for the native Mail app in IOS does not work. I almost fractured my wrist trying that with Outlook.
1. Paste formatted signature in the left box of http://wordtohtml.net. You can compose your signature in Word.
2. Select any tags you want removed from the checkboxes below the left box and click on the “Cleanup my HTML” button.
3. Copy the html generated in the right window.
4. Send an email to yourself with the only the html signature in the body of your email. Remove any signatures that are already there.
5. Open the email in Outlook app on your iOS device. Copy the HTML signature.
6. Go to Settings (gear icon) in Outlook app and then to the signatures section.
7. Paste the HTML signature there.
8. Send test email.
You can use any HTML editor to format your signature. My steps above are for those who do not have such tools.