Passive of "show off"

The use of "showed" is rare (OALD, ngram).

There is no problem with the passive, there are plenty of examples here.

However, the meaning is not "boast" but "to show people something or somebody that you are proud of".

The problem with your sentence is not that you are using show off in the passive, but that the subject of the sentence doesn't work with the passive of show off because it wouldn't be the object of show off if the sentence was in the active voice.

Your sentence is:

I wonder if she had been the same as me, always being showed off by that genius.

Putting this back into the active case, you get

that genius was always showing her off,

which would make sense if she was the genius's wife or daughter, and he was boasting about her. However, from context it doesn't seem that this is what you mean.

I am assuming that what you really want to say was

that genius was always showing off to her.

The only way to put this in the passive would be

always being shown off to by that genius,

which, if it isn't ungrammatical¹, is at least quite awkward; I would rephrase the original sentence.

¹ Putting verb/preposition pairings into the passive is a complicated topic—sometimes you can do it, and sometimes you can't, and I don't know the rules.

Without more context, it sounds like the author is describing an implied victim of someone "show[ing] off".

If this is the case, a better phrase is to "show up" as FumbleFingers commented above:

2: to embarrass or cause to look bad especially by comparison

trying to show up the boss

Those compared to the show-off would be shown up.