Q: Inconsistent iMessage behavior between OS X and iOS

Your settings might be different in iOS and OS X.

1. Check your settings/preferences. Especially for Message History and Expiration options.

2. See number 1 above. This is related to Message History options.

Although they seem to be the same App, they are not.

The OSX Messages app does not appear to have settings for message retention.

There is no iCloud element for Messages. Messages does not sync to iCloud.

It's all local.

I suppose you could make a copy of the local Messages data and keep it in the iCloud Drive folder. Not exactly automatic, but you'd have some level of data backup.

This a feature called Continuity.

Use Continuity to connect your iPhone, iPad, iPod touch, and Mac - Apple Support

It's done through the Messages app, not any iCloud feature.

Actually... Instant HotSpot, Handoff and Cellular Calls require the iCloud account... Continuity is 4 different features, each with different requirements. None of the features use iCloud for storage.

SMS uses Messages, Cellular Call uses Facetime & your iCloud ID, Handoff needs your Apple ID, Bluetooth & the same Wifi network to be on, and HotSpot uses iCloud & Apple ID.