The multi classing rules state (PHB p164):
If you already have the Channel Divinity feature and gain a level in a class that also grants the feature, you gain the Channel Divinity effects granted by that class, but getting the feature again doesn’t give you an additional use of
Therefore, RAW, the character gets both the Channel Arcana uses from the Wizard class and the Channel Divinity uses from the Cleric class as they are different class features. Even though Channel Arcana allows the use of a channel divinity option it is not Channel Divinity itself.
This is clearly very powerful, a work-around the multi-class rules put in place to limit the number of channel divinity options that can be used by a single character between short rests.
The obvious house rule is to treat them both as Channel Divinity for uses under the multi-class rules, even though Channel Arcana also allows Divine Arcana which is not a technically a Channel Divinity option.
I have no experience of running such a character so this is all theoretical.