That's not a straightforward reading of the relevant passages on the spell casting bonus.
At first, I was just thinking that proficiency with a magical focus should let you add your proficiency bonus, because that's simple and this game is simple. It's the same logic for why a mage can cast in armor in the first place.
Then I was thinking like you suggest, and like all of the previous editions implied - that mage spells are inherently different from cleric spells, are powered by different stats, and nothing translates between them.
On third thought, though, I think the first idea makes more sense. Consider it like mage spells and cleric spells are two ways of tapping into the same type of thing. Whether you use a rod or a holy symbol, you're doing the same things with it. Why else would a cleric 1 / mage 19 be able to cast
cure wounds IX ? It must be the case that
most spellcasting experience translates easily between different classes.
If you take it that way, then you can use any magical focus that you know how to use, in order to cast any spell you know from any class (using your universal spell slots). If you read the part about save DCs, in that light, then you can use any one of your ability scores from any of your casting classes in order to determine your save DCs. Kind of like how a multi-class monk / barbarian can use either Con or Wisdom in determining AC.
It may not make a lot of sense from the way any of the previous editions did things, but it fits in pretty well with the way 5E has been presenting itself.