I would say that each of your three sentences conveys a slightly different meaning:
"I suppose it is made up of some sort of tissue that is getting harder as the animal gets older."
Here, "as the animal gets older" is a natural occurrence and time frame, but is contrasted to something that the speaker supposes to be occurring gradually and perhaps unnaturally (e.g., an illness). The continuous conjugation "is getting harder" allows for multiply interpretations and awaits clarification in a later statement.
"I suppose it is made up of some sort of tissue that gets harder as the animal gets older."
The simple conjugation of each verb expresses what the speaker supposes to be a fact and, for this animal, a fact of life: as it gets older, the tissue gets harder.
"I suppose it is made up of some sort of tissue that is getting harder as the animal is getting older."
Is the first sentence wrong or unnatural, or am I just imagining things?
Unnatural, yes...wrong, not necessarily. Once again, the continuous conjugations beg questions and await clarification in later statements. Why say "as the animal is getting older"?
Were the clause preceded by a comma, "as" would be synonymous to "because" and aging would be the supposed cause of the hardening. This would be a justified and grammatically sound reason to say "is getting". Perhaps this is just a typo or lazy punctuation.
Another possibility (no comma) is that the hardening has not only been occurring gradually but has also been occurring for quite some time, or that its consequences are worsening or becoming more problematic as the animal ages. It could also be that the speaker (vet?) wishes to suggest that the hardening will only get worse as the animal continues to age, and so on.
The conjugation is a bit odd but "is getting older" may indeed serve a purpose, depending on the intention of the speaker and surrounding context.