Learn Danish: Hver stemme tæller.

Yes! In Danish, en stemme means both a voice and a vote. The verb at stemme also means to vote. You will often see this double meaning in political contexts.

Could I say alle stemmer tæller instead of hver stemme tæller?

Yes, alle stemmer tæller (all votes count) is grammatically correct and carries the same general meaning. However, hver stemme tæller emphasizes the individual importance of each single vote, which makes it the standard idiomatic phrase used during elections.

Does tæller genuinely mean "matters" here, or is this just an English idiom translated directly?

It genuinely means matters or has significance in Danish too. Just like in English, the verb at tælle (to count) works both literally, as in counting numbers, and figuratively, as in being important.

Why is it hver and not hvert here?

Hver is used because stemme is a common-gender noun (en stemme). If it were a neuter noun, like valg (election), you would use hvert instead, such as in the phrase hvert valg.