... he set two objectives/objects.

[WilliamTaft]

Source: The Russian Empire - Summary on a map, YouTube

Michael I is the first of a new, very long dynasty. But he inherits a devastated country, ruined by years of wars and instability. Groups of looters plunder the countryside, while Polish and Swedish armies still occupy Russian territories. In order to restore the state, he set two objectives: to ensure peace, and to stabilize the state’s finances.

Can I say objects in this context? If so, what's the difference?

 

[5jj]

Can I say objects in this context?

No.

 

[teechar]

Can I say objects in this context? If so, what's the difference?

No! These words have different meanings in English.
objective = goal/target.
object = thing/item.

 

[WilliamTaft]

No! These words have different meanings in English.
objective = goal/target.
object = thing/item.

Thank you, teechar.
In several dictionaries it do have the meaning of goal, purpose, aim, target etc. Is it seldom used in modern English?
These are from the Britannica and Longman Contemporary:

 

[jutfrank]

I think it's very hard and barely worthwhile attempting to outline any semantic differences between the two words in this particular sense of a goal. Instead, I'd focus on any idiomatic or grammatical differences, i.e., differences in use rather than meaning.

For example, in the pattern 'the _______ of', it will be likely that 'object' will be used. And in the particular context of post #1, given that the word is partnered with the associated collocate verb 'set', only 'objective' fits.