Before spontaneous symmetry breaking (SSB) elementary particles belonging to the same $SU(2)$ doublets are indistinguishable, which clearly is not the case after SSB.
I am comfortable with the idea of mass generation from the Higgs mechanism, but I was wondering how does this mechanism lead to different masses for the same doublet?
At which stage of the mechanism is this difference introduced, and would it be physically consistent to imagine a SSB with same masses for the doublet elements?
Decent Standard Model textbooks would make this question impossible. E.g., M Schwartz's, Sec. 29.3.2, Cheng & Li, sec 11.2, etc. Unfortunately, condensed texts and complacent sources introduce the SM without building the background for the various pieces that fit, sudoku like, together, uniquely and superbly elegantly, and throw a jumble of half-digested words to the student.
The Higgs mechanism, strictly speaking, does not give mass to the fermions, only the (formerly) gauge bosons. What makes the Higgs mechanism possible is SSB, which can exist and luxuriate outside the ambit of gauge symmetries, and also, logically independently, gives fermions masses: the second job of the Higgs.
E.g., in the σ-model for global chiral symmetry breaking, a prerequisite for any introduction of the SM; many authors assume students learn all this in chiral model preambles. The generation shaping the SM knew that inside-out, and did not emphasized it in its early summaries (and subsequent introductions failed their pedagogical mission on this, sometimes introducing SSB through the SM, a laughably terrible idea...). Moreover, a crucial piece of the fermion mass mechanism is the SU(2) group theory of the conjugate representation. (This stood in the way of the community accepting that neutrino masses were a logical part of the SM structure...)
Before spontaneous symmetry breaking (SSB), elementary particles belonging to the same (2) doublets are indistinguishable, which clearly is not the case after SSB.
Not quite: The $SU(2)\times U(1)$-invariant Lagrangian also has Yukawa couplings, which appear, in anticipation of SSB, to be treating us and ds differently. (Stick to only one generation to avoid extraneous complications due to flavor mixing.)
$$
{\cal L}_Y= -y^d \bar Q H d_R -y^u \bar Q \tilde H u_R,
$$
where Q is the left-handed $(u_L,d_L)^T$ doublet you have been mostly focussing on, the R-fermions are SU(2) singlets with suitable perverse (!) weak hypercharges, and the arbitrary Yukawa couplings are end-reality-driven $y^d\propto m_d/v,~~ y^u \propto m_u/v$.
This addresses your final question: these two might be the same, as per your fantasy, at tree level; but you never know what radiative corrections involving their different hypercharges might end up inducing. This would be an accidental degeneracy broken by their explicit breaking!
Now for the interesting part. As linked, the doublets $\tilde H\equiv i\tau_2 H^*$ and H transform identically under SU(2), but oppositely under the hypercharge U(1), and their v.e.v.s end up distinctly different, $$ \langle H\rangle_0= (0,v)^T,\\ \langle \tilde H\rangle_0= (v,0)^T, $$ so they select very different pieces of the SU(2) dot products, resulting in different masses, $m_d$ and $m_u$, respectively, $$ \langle {\cal L}_Y \rangle_0\propto -y^d(\bar u _L ~~~\bar d_L)\left(\begin{array}{c} 0\\ v\end {array}\right)d_R -y^u(\bar u _L ~~~\bar d_L)\left(\begin{array}{c} v\\ 0\end {array}\right)u_R . $$ This has nothing to do with gauge couplings, or the Higgs mechanism: it's pure global SSB.
In a way, this is the most interesting bit of flavor physics, and the trickiest sudoku piece to fit.