The top number - in red for the image below - is how many beats are in a measure (or bar.)
The measure are indicated on the staff with vertical lines - measure are shown in the image below with a red bracket.
The bottom number is which note is used for the beat in this example the 4 means 1/4 (quarter) note. Notice that each quarter note is numbered
1, 2, 3... to show the counting of the beats. 3 beats for 3/4 and 4 beats for 4/4.
A nice PDF about meter is available from Toby Rush.
With that covered...
Is the measure the second 4 that tells us it is 1/4 of the duration of
No. Now we know that there are 3 or 4 beats in the beats depending on the meter.
Would it be then correct to say that a time signature of 4/4 with
90bpm would result in a single note duration of 0,17s and that there
need to be at least 4 notes that equal 4 quarters?
No. 90 bpm / 60 secs = each beat is 0.66 seconds. 0.66 sec for quarter notes, 0.33 sec for eighth notes, etc. If you switch to 3/4 meter, these times per rhythmic note value will stay the same. But the feel of the phrases will shift from even 2's and 4' to 3's. Like the feel of a polka versus a waltz.
I think there may be some confusion about bpm/tempo and meter, something like a bpm number is a sort of meter.
Perhaps the thing that needs to be said is meter is about placement of accents.
Let's say you had a bpm/tempo of 120 bmp and the beginning of the music was just a kick bass drum playing quarter notes, just playing the beat. Importantly, imagine that the drummer (drum machine?) didn't place any accents on the drum. How would we hear that the music was in 3 or 4? We wouldn't know what the meter is.
We need something to happen to create an accent. The basic theory is that an accent will go on beat 1. That can be accomplished many ways. You could simply hit the kick louder on beat 1 and light on beats 2, 3, or 4. You could have another drum or instrument play on beat 1. You could use a pick up note like...
|1 2 3 4 +|1 2 3 4 +|1
All digital recording programs I have seen include a metronome/click track with a bell feature that will play some bell or other accent on beat one. That bell is there to help hear the meter more clearly.
So, let's image this kick bass plays a pickup note and stomps on beat one hard followed by 3 soft beats. That's a total of 4 beats with an emphasis after the pickup note that let's us hear that the kick after the pickup is beat 1. If we are familiar with sheet music, we can imagine that beat 1 coming after the barline.
If we speed up the bpm to 145, the measure do not get shorter. Each measure will be the pattern of one accented beat followed by 3 softer beats. The measure is still 4 beats. The duration of actual time will shorten, but the measure length - which is measured in beats rather that duration in seconds - stays the same.
Maybe the important distinction to make is between meter length and duration in seconds.
- meter length is measured in beats per measure with measure delimited with an accent on beat 1
- beats per minute (bpm) is the duration in seconds of a beat