I've checked the specific conditions for the MW test to be a test of medians and its o.
The Mann-Whitney test is
not at test of medians.
Read about its properties and search for publications by Fagerland Sandvik, like
in this one or
in this one.
The histograms are have more or less the same shape, but i would like to do a t-test if the results are valid and interpretable offcourse.
Besides doesnt he CLM theorem declares that if your sample is sufficiently lare enough you can always use a t-test?
Yes, the t-test is robust to deviations from the usual assumption (symmetry and constant variance) but it is not robust to outliers. And from the histograms it looks like there are outliers in the upper and lower ends of the scale. These will be very influential observations.
The first histogram looks very skewed. Maybe by taking logs (several times) or the square root (several times) it can be made more symmetric. The last one looks more like double exponential (Laplace distribution) than normal. But even there there are some outliers.
Otherwise I agree with the OP that CLT would take care of the problems but the outliers make me hesitate.
But, if the purpose is to run a logistic model and use these variables as explanatory variables then it does not matter. There are no distributional assumptions on the x-variabels i regression model, logistic or not.