Why does a synchrosqueezed wavelet transform show oscillating behavior?
This was interesting to figure out. The key lies in the phase transform, and how CWT interacts with own derivative upon insufficient component separation. Relevant are, and I'll be answering, the following:

What causes the wavy pattern?
Is the wavy pattern truly a sine, or a lookalike?
Why is only one band wavy, not both?
What can be done about it?

Answer split in two to separate meat from supplement, but much of insight's in latter; sections labeled in recommended reading order.
TL;DR: skip to Conclusion. Answer code.

0. How does the phase transform work?
A helpful requisite for following sections, covered here.

1. What causes the wavy pattern?
Begin by visualizing the phase transform, w:

The lower frequency's CWT and SSQ are nearly that of a pure tone, but the higher frequency is all wiggly. Let us plot one such wiggly row:

This sure looks sinusoidal. But is it? Stay tuned.
First: is the wavy SSQ explained by the w? Absolutely: w is wavy across timesteps, meaning for every column, we reassign CWT rows to a different SSQ row, in a wavy/sinusoidal pattern (see [complement] on reassignment). Our answer thus lies in understanding why w is wavy to begin with. Second answer begins here.

4. Inseparability: pre-transform perspective
Recall, the CWT and dCWT operations both take place in the frequency domain; we reach the lowest level of analysis by inspecting it directly. With xh = fft(pad(x)), psih = freq-domain wavelet, and dpsih = freq-domain wavelet with the derivative (* 1j * xi / dt), we plot both, overlapped and rescaled to fit in same graph, as well as the result of the multiplication - for our same row, and one on higher scale:

First, note for our original row scales[80] = 5.08, we indeed have a sum of two pure cisoids (analytic wavelet -&gt; all negative frequencies are zero, and Morlet has pure-real DFT; cosine input -&gt; pure real DFT; thus a DFT coefficient multiplies its complex exponential basis without cancellation), one at higher frequency having much greater amplitude.
Next, compare the cisoid amplitudes for psih vs dpsih; higher frequency dominates more for dpsih than for psih for both example scales. Coincidence? Well, they both do * xh, thus we expect the dpsih wavelet itself to skew toward right. Plotting,

this is confirmed. Does this happen for every scale? Yes: * xi = * linspace(0, pi, N) - taking the derivative involves multiplying by a line, thus skewing the entire wavelet toward higher frequencies, also breaking even symmetry (though not a lot).
All puzzle pieces are in. The skewness creates a difference in relative amplitudes at the two frequencies w.r.t. the non-derivative. The general form for dWx / Wx is then (still assuming purely real fft(Wx)):
$$
\frac{dW_x}{W_x} = \frac{a\exp{(j(\phi_1 + \pi/2))} + b\exp{(j(\phi_2 + \pi/2))}}{c\exp{(j\phi_1)} + d\exp{(j\phi_2)}}, \tag{4} \\
\phi_i = \omega_i \frac{n}{N}, \ \ n=[0, 1, ..., N - 1]
$$
($+\pi/2$ per the derivative). Let's first inspect interesting cases numerically, then derive an exact relation as proof (second answer).

8. What can be done about it?
Strictly speaking, nothing, unless the wavelet decays completely to zero eventually (untrue for many wavelets); as long as there's more than one frequency in the entire DFT of the input, there will be some form of interference. However, practically, such interference can be made negligible by:

Ensuring bands are sufficiently separated
Reducing wavelet frequency width (for Morlet, increase mu)

Let's inspect Morlet mu=5 vs mu=20:

mu=20 is clearly narrower; to get a better idea, single out a few rows:

Overlapped at same center frequency (~same row):

We thus predict mu=20 will show no wiggliness we can see. In fact, mu=20 is overkill, we'll settle for mu=10:

Can we also guess how to make both bands wiggly for small mu? Note, increasing scales lowers the wavelet's center frequency (shifts it left), but also reduces its frequency width; thus our wiggle odds improve toward lower scales (or higher frequencies). This turned out easier said than done; after some experimentation, I've had limited success, but maybe it can be done better:


9. Conclusion

What causes the wavy pattern? The frequency-domain wavelet and its derivative intercept the input bands at slightly different ratios per latter being skewed. The resulting phase transform is (imag. part of) a ratio of sums of complex sinusoids, which is of form $(\cos(\phi) + k_1)/(\cos(\phi) + k_2)$. The reassignment pattern dictated by w can then have a sine-like pattern across rows, which manifests as a sine-looking synchrosqueezed CWT.
Is the wavy pattern truly a sine, or a lookalike? Latter; it can never be exactly sine, but close to. The pattern is characterized by $(\cos(f_1 - f_2) + k_1)/(\cos(f_1 - f_2) + k_2)$, and has a period of $|f_1 - f_2|^{-1}$, where $f_1,f_2=$ input frequencies.
Why is only one band wavy, not both? With increasing scale (and decreasing center frequency), the frequency-domain wavelet moves to the left while shrinking in width; the resulting cisoid proportions work out such that dominantly sine reassignment pattern is toward the higher input frequency, where the wavelet is wider, but the lower one isn't devoid of such a pattern either.
What can be done about it? Separate input frequencies more, or reduce wavelet frequency width across all scales (for Morlet wavelet, raise mu).