"It is 8:08."
More briefly; no.
It has only to do with typing the character. Its a quirk that helps when a word ends with ん, and the next begins with a vowel, in which case you dont want them to combine.
You can also notice that if you type ん with a single n followed by a consonant, it will automatically type ん without needing the second press. So it depends on what comes after.
Two kanji used together (in a word, or a construction like the one here) are very likely to be read completely differently then when used in separately. So learning the pronounciation from such exercises as this one introduces very bad habits. I wonder how difficult would it be to create separate recordings for kanji blocks which are meant to be used in a single word / construction. Also, it would be nice to have some kana added to the translation for people who are not able to listen.
In case anyone needs a refresher on how to pronounce the minutes: 一分 (-:01) | いっぷん or ippun ニ分 (-:02) | にふん or nifun 三分 (-:03) | さんぷん or sanpun 四分 (-:04) | よんぷん or yonpun 五分 (-:05) | ごふん or gofun 六分 (-:06) | ろっぷん or roppun 七分 (-:07) | ななふん or nanafun 八分 (-:08) | はっぷん or happun 九分 (-:09) | きゅうふん or kyuufun 十分 (-:10) | じゅうぷん or juufun