There are many cases in which multiclassing spell slots can create unclear areas in the rules, but I don’t think this is one. The only requirement from the manual of golems is that you have at least two slots.
For multiclassing spellcasters slots are determined by your combined spellcaster level and do not belong to any one class.
Spell Slots. You determine your available spell slots by adding together all your levels in (spellcasting classes). Use this total to determine your spell slots by consulting the Multiclass Spellcaster table.
Looking at the table referenced, you see that a wizard 5/cleric 5 does indeed have two available 5th level slots. There is no differentiation in the rules that one of those slots is a wizard slot and one is a cleric slot. You can use those 5th level slots to cast any spell you know from either class.
Seeing that we do not need to differentiate where those slots came from (and the rules don’t differentiate them anyways), they would meet the manual of golems restriction with no caveats.
One confusing point might be that multiclassed spellcasters do indeed use their individual classes for one aspect in particular: determining what spells they know and/or can prepare.
Spells Known and Prepared. You determine what spells you know and can prepare for each class individually, as if you were a single-classed member of that class.
However, this is only the rule for determining what spells you know not how many.
So, while it is true that the cleric/wizard does not actually know/cannot prepare any 5th level spells (only upcast from lower levels), that should not matter in this case.
The manual of golems does not require a character to use or expend any 5th level spells or even spell slots to use the item. It only requires that they have them. And, as shown above, they definitely have the required number of slots.
The spell slot restriction on the manual of golems seems to be a restriction put in place to ensure that only powerful spellcasters can use it. Allowing multiclassed spellcasters to use it with combined slots not only follows the rules but also still abides by the presumed intent of the spell. Even though their spellcasting prowess is split between more than one discipline, they are still a very powerful spellcaster.
From a balance/rules perspective, multiclassing will never allow a character to be able to use this item before a single-classed full spellcaster would. In fact, many combinations will actually slow down spell progression meaning the multiclassed character would have to be a higher level in order to use it. This seems very straightforward, well within the rules, and fair.