Language models (LMs) are known to represent the perspectives of some social groups better than others, which may impact their performance, especially on subjective tasks such as content moderation and hate speech detection. To explore how LMs represent different perspectives, existing research focused on positional alignment, i.e., how closely the models mimic the opinions and stances of different groups, e.g., liberals or conservatives. However, human communication also encompasses emotional and moral dimensions. We define the problem of affective alignment, which measures how LMs' emotional and moral tone represents those of different groups. By comparing the affect of responses generated by 36 LMs to the affect of Twitter messages, we observe significant misalignment of LMs with both ideological groups. This misalignment is larger than the partisan divide in the U.S. Even after steering the LMs towards specific ideological perspectives, the misalignment and liberal tendencies of the model persist, suggesting a systemic bias within LMs.