Why would a Japanese adult be embarrassed to use the word “friends”?

Question

I am watching a current anime, That Time I Got Reincarnated as a Slime ( 転生したらスライムだった件 ). The protagonist, now a reincarnated slime in the isekai story, is making friends with a dragon. The specific conversation begins at time code 20:30 of chapter 1*. After they have agreed to be friends, he thinks to himself (sub, time code 21:17):

いい年して『友達』って、ちょっと照れるけど…。
It's a little embarrassing talking about "friends" at my age...

He was 37 when murdered. Why would he be embarrassed to use the word "friends"? Is there a better word to be used between adults?

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_{* Link is to Crunchyroll, an American streaming service. I don't know what other countries will be able to view it.}

_{This question is a follow-up to What is Japanese version of the pun word, “slife”? … and term for adult friend?. On Anime & Manga SE. @кяαzєя answered it, and he suggested I post a follow-up question here.}

Answer 1

いい年して「友達【ともだち】」って、ちょっと照れるけど。

The word tomodachi ("friend") itself is not really embarrassing, although there is a more formal word for this concept. In this case, this tomodachi also represents the whole exchange he has just made with the tsundere dragon. Indeed, innocently and directly saying 友達にならないか ("Let's be friends!") is not something a typical middle-aged man would do, at least in Japan. It sounds to me like what a kindergartner might say. Usually such a relationship between adults starts without mentioning the word tomodachi. (By the way, the situation is different when it comes to romantic relationship.)