Coef not defined because of similarity and pvalues

I have a CRM data set used for an experiment, where the dummy W corresponds to the treatment/control group (see code below). When I tested for the independence of W from the other features, I realized two things:

• When using model.matrix, some coefficients (1 in this dummy dataset) were not defined because of similarity. This did not happen when feeding the DT straight to lm()
• The model obtained in both cases produces different results i.e., the p-values of the individual features changes

I (think that I) understand the concept of multi-collinearity but in this particular case I don't quite understand a) why it comes up b) why it has a different impact on model.matrix and lm

What am I missing?

Thanks a lot!

set.seed(1)

n = 302

DT = data.table(
    zipcode = factor(sample(seq(1,52), n, replace=TRUE)),
    gender = factor(sample(c("M","F"), n, replace=TRUE)),
    age = sample(seq(1,95), n, replace=TRUE),
    days_since_last_purchase = sample(seq(1,259), n, replace=TRUE),
    W = sample(c(0,1), n, replace=TRUE)
)

summary(DT)

m = model.matrix(W ~ . +0, DT)
f1 = lm(DT$W ~ m) 
f2= lm(W~ ., DT)

p_value_ratio <- function(lm)
{
    summary_randomization = summary(lm)
    p_values_randomization = summary_randomization$coefficients[, 4] 
    L = length(p_values_randomization)
    return(sum(p_values_randomization <= 0.05)/(L-1))
}

all.equal(p_value_ratio(f1), p_value_ratio(f2))

alias(f1)
alias(f2)