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I have a data set with gene expression levels measured for three different treatments, and for each treatment we have three biological replicates. Treatments are: Control, 24h at $4$°C, and 24h at $-3$°C. This analysis was carried out on two genotypes.

                       CTRL1 CTRL2 CTRL3 4°C_1 4°C_2 4°C_3 -3°C_1 -3°C_2 -3°C_3
Gene1  Genotype 1       20    22    25     85    92    120   450    380    372
       Genotype 2       50    72    78     130   122   184   250    490    700

I'd like to know how to compare gene1 expression level in genotype1 vs. genotype2. The aim is to find a similarity between expression trend in genotype1 and genotype2.

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Assuming you have more data than you have shown, a two-way ANOVA with Genotype, treatment and their interaction would give you a hint. If the interaction term is not significant this implies there is no difference in how the genotypes are responding across treatments.

If this is all the data, you have very little power to detect significance. The best you might do is plot the data and eyeball it. If the genotypes go up and down together between treatments, it suggests no difference in how the genotypes are responding across treatments (same as the ANOVA interaction term). The visualization might work better if you normalize the data so they are on the same scale.

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