When matching an egg donor to a recipient, fertility clinics take into account that there is compatibility between them and that they share as many phenotype and immunological characteristics as possible. So, theoretically, yes, a baby conceived with donor eggs is likely to look like the birth mother.
However, it is important to note that the biological mother of the baby will be the egg donor, so in terms of genetic inheritance, the answer to this questions is no, the baby will not look like the recipient, since he or she does not share the genetic load with her.
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