Sometimes a quantile isn’t uniquely defined (often solved by taking the average of the endpoint of the interval of quantile points). However, it only makes sense to use a definition that ensures that median and 0.5 quantile are identical.
In this case, it seems that things are worse. To me, it seems that the result of quantile is just wrong. The 0.5 quantile and the median, say m, is the same thing and should satisfy P(X \geqslant m) \geqslant \frac{1}{2} and P(X \leqslant m) \geqslant \frac{1}{2}. For your inputs, I get
julia> x = [1; 4; 3; 2; 2.5; 7];
julia> w = [0.1;0.3;0.05;0.05;0.2;0.3];
julia> sum(w[x .<= 3.5])
0.4
so 3.5 is not a median. To see that 4 is the unique median, you can create the following table and see that the row with x=4 is the only one that has probabilities higher than \frac{1}{2}.
julia> p = sortperm(x);
julia> table(cumsum(w[p]), reverse(cumsum(reverse(w[p]))), x[p], names = [Symbol("P(X<=x)"), Symbol("P(X>=x)"), :x])
Table with 6 rows, 3 columns:
P(X<=x) P(X>=x) x
─────────────────────
0.1 1.0 1.0
0.15 0.9 2.0
0.35 0.85 2.5
0.4 0.65 3.0
0.7 0.6 4.0
1.0 0.3 7.0