I find friendship and romantic relationship to be very close to one another. If I think more about it, one difference I find is that in a romantic relationship the tension between the similarity and the difference between oneself and the lover is the thing that attracts one most. I would like to emphasize the difference, if there is too much similarity then it does not work. Maybe the reason might be that one would like to understand another person, so do not want to see a copy of oneself. 
Whereas in a friendship complete similarity of the souls would be the ideal thing, just like the case when Montaigne speaks of La Boetie. In that kind of a relationship I guess one wants their ideas, prejudices etc. to be confirmed. So maybe in a sense a friendship is more vain in this respect. I cannot not quote Shakespeare's Sonnet 30 though, which, for me, summarizes the best aspects of true friendship: 

When to the sessions of sweet silent thought
I summon up remembrance of things past,
I sigh the lack of many a thing I sought,
And with old woes new wail my dear time's waste:
Then can I drown an eye, unused to flow,
For precious friends hid in death's dateless night,
And weep afresh love's long since cancelled woe,
And moan the expense of many a vanished sight:
Then can I grieve at grievances foregone,
And heavily from woe to woe tell o'er
The sad account of fore-bemoaned moan,
Which I new pay as if not paid before.
But if the while I think on thee, dear friend,
All losses are restor'd and sorrows end. 

An important question Edna.
You say: "Even more so when it comes to women where the relationship of friendship seems to many an afterthought of the important relationship – the erotic one."
I think part of the reason this is the case is that it's less clear what is being produced by women-friendship, while we do understand what is being produced by the erotic relationship - kids and family. In the case of men, friendship is historically very important in battle, in politics, in everyday life. Of course what is being produced by friendship is yourself, but that's harder for others to see. Our culture puts much more emphasis, especially for women, of creating kids than of creating ourselves; maintaining, of course, that kids will create you but also seeing them as the only way to create yourself. (Parents want their grandchildren).
 
I found your example of trust an excellent one, but we should notice a change from Montaigne's time. While Montaigne had a strictly romantic relationship with his wife the romantic relationship today is often a mix of friendship and the erotic. I think that erotic relationship, in their essence, are lacking in the element of trust, an element which exists in abundance in true friendship. So I think people today may still be suspicious when it comes to love matters, but are trusting of their spouse as a friend. This is at least the case in many relationships.

Que sais-je, I hope you don't mind I call you Q as I can't speak to someone with your name. Q, very interesting distinction. Personally, I'm not sure to which direction it should go. I think some people would want in the romantic relationship a complete similarity of the souls and a difference when it comes to friends, and for some it's vice versa, but it's an interesting distinction. It also matters what we would call "similarity of the souls." I want both my romantic relations and my friendships to be with people who are good in nature, nice people, etc. I want a specific kind of soul, but then what kind of shape it takes in the world matters much less.
I do though know people where all of their friends are in their own profession (in a general sense) and that's the only people they are interested in talking with.


Here's a famous passage from the bible, which I had to learn by heart in school (The second book of Samuel, chapter 1, King James translation). David lamenting the death of his friend Jonathan.

David's Lament over Saul and Jonathan
17  ¶ And David lamented with this lamentation over Saul and over Jonathan his son:
18  (also he bade them teach the children of Judah the use of the bow: behold, it is written in the book of Jasher: Josh. 10.13 )
19  The beauty of Israel is slain upon thy high places: how are the mighty fallen!
20  Tell it not in Gath, publish it not in the streets of As'kelon; lest the daughters of the Philistines rejoice, lest the daughters of the uncircumcised triumph.
21  Ye mountains of Gilbo'a, let there be no dew, neither let there be rain, upon you, nor fields of offerings: for there the shield of the mighty is vilely cast away, the shield of Saul, as though he had not been anointed with oil.
22  From the blood of the slain, from the fat of the mighty, the bow of Jonathan turned not back, and the sword of Saul returned not empty.
23  Saul and Jonathan were lovely and pleasant in their lives, and in their death they were not divided: they were swifter than eagles, they were stronger than lions.
24  Ye daughters of Israel, weep over Saul, who clothed you in scarlet, with other delights; who put on ornaments of gold upon your apparel.
25  How are the mighty fallen in the midst of the battle! O Jonathan, thou wast slain in thine high places.
26  I am distressed for thee, my brother Jonathan: very pleasant hast thou been unto me: thy love to me was wonderful, passing the love of women.
27  How are the mighty fallen, and the weapons of war perished!

A connection which is important to note, when thinking about friendship compared to other relationships, is that of friendship and conversation.
While at first glance the strong linking of the two isn't as clear, it becomes clearer when one starts to think about it. For example, Proust was against both. In friendship we seek conversation, and in conversation friendship, while in the erotic relationship something different is aspired to. Sex is clearly important, but it is somehow more connected to theater and acting than to dialogue - and yes, even great sex. (Perhaps connected to a conquering.) While friendship is about a conversation of equals. And a serious conversation infuses a feeling of friendship.