Both are grammatically correct.
In the former, 声がします expresses/implies that someone's voice came toward the main character (浦島太郎 here) and he hears it.
So the latter is just:
Someone called, "Urashima-san, Urashima-san..."
while the original sounds more like:
Taro heard someone's ...
あげる is an ichidan verb (あげる、あげます、あげた...). On Jisho, "to give" is definition 18; when the verb is used to mean "to give," it's almost never written with kanji.
I'm not sure where you got the idea that あげる is colloquial. It's common and appropriate in speech and writing in nearly all contexts. (A couple of exceptions: when you need to reach for a humble verb, ...
To use punctuations for you, which manga will never do:
The part of your question that worries me the most is where you said:
"Maybe here it simply means "without discriminating him ...
No, there is no rule that you always have to use the passive form.
However, I realized that a Japanese sentence often omits the subject, often implying the subject is the speaker without other context, and that this style makes the sentence in the passive form.
He told me what I needed to know.
can be translated to
This is a combination of three grammar points, namely も, -てくる and sentence-end te-form.
It started to snow.
This てくる describes something is coming toward you, mentally, temporally or physically. Difference between -ていく and -てくる
It started to snow, and/so ...
This te-form is where "now that" came in. See: て form at end ...
This 乗り切る is not "to overcome", but a simple combination of 乗る + 切る, "(for power/force/etc) to be put/applied/loaded" + "completely/fully". 荷重が乗り切ってから is "after his weight was fully put onto the punch". It can be rephrased as 荷重を乗せ切ってから ("after loading/putting his weight fully onto the punch").
So, in this case 呼ぶ声 is meant to be taken as a sort-of participle which means "a calling voice"/"a voice calling out". So the basic translation of what the sentence is - "'Urashima-san, Urashima-san', someone's voice called." You could use just "呼びます", the difference is mostly a stylistic one. Japanese uses "がします/する" instead of the literal verb, "call" in ...