I think in general what you are looking for doesn't tend to exist (with exceptions, such as those in other answers). As background, I am a theoretical physicist with what I suspect is a stronger mathematics background than most, but I think theoretical physics (and probably computer science) is the only field where you could really expect to find books that satisfy:
Strong preference would be to concise, terse texts that are foundational but totally rigorous.
To find stuff like that you'd probably have to look at a very small selection of research papers (some of which I'll list at the end of this answer). However since these are research papers they tend to be focused on the subfield and probably assume some familiarity already (kind of defeating the purpose to learn from them).
I would say that the lack of these textbooks is not a bad thing, and that's because mathematics tends to be different to basically any other science. In mathematics you can start with your axioms and prove what you would find if you did experiments with that mathematical structure. While in basically every other field you start with knowing the results of experiments and have to try to deduce which axioms you should start with. There are probably exceptions to this but I think in general this argument holds.
To give a bit of a feel why this is different something I remember hearing about the theory of (I think) topologically ordered phases a bit over 5 years ago: when you read papers there are maybe 3 different ways that people define these phases, which aren't equivalent but kind of do the same thing in the end. While this might sound like there should be 3 different definitions used and introduced at the beginning of each paper, the end goal is to discuss and explain real experiments and so we only want and we only expect a single theory to exist. The different ways people have defined things the definition becomes a "I know it when I see it" situation and everyone working on the topic, even if using different definitions, should expect to see the same kinds of behaviours.
The other thing to keep in mind is that if you have a rigourous theory that uses sophisticated mathematics compared to a simple theory that only requires arithmetic, if the simple theory matches the experimental data better then the rigourous theory then that's the one that should be kept since the goal is to explain these experiments. Also if you can explain the same set of experiments with either simple or sophisticated mathematics with the only difference being rigour, unless your sophisticated mathematics approach is more powerful in explaining or making predictions for other experiments that the simple approach cannot, then the simple approach will remain dominate just because of the cognitive overhead associated with using more sophisticated mathematics.
The final thing is to think about is where would you start if you wanted to give a foundational description of genetics? we could start with central dogma of biology which is that DNA -> RNA -> proteins, explain what all these things are and build up how proteins effect the behaviour of cells and eventually an animal along with expressions of phenotypes (what it looks like). From this construction you could derive evolution. The problem with this approach is that we can't do it yet, there are heaps of things we don't understand about proteins and protein folding (which is important for how a protein changes cell behaviour), and when do you introduce the ability to inhibit/enhance different gene expressions (protein production) via the existence of certain proteins. This is a more complicated topic that if introduced too early distracts the reader from the core idea of DNA gives proteins which give looks/behaviour.
Instead it makes more sense to start with talking about the looks and behaviours you see and saying it is conjectured to have been caused by ... which is conjectured to be caused by ... and so on. This also means that new research, which is more likely to be at the molecular scale than animal scale, is only a correction to the end of your book, rather then the very first sentence. This would be different to a maths text where new research won't overturn the axioms but would be more likely to add new knowledge about the structure that arises from these axioms.
We could discuss the looks and behaviour in terms of other mathematical structures to give them a rigourous definition, but WHY? You add cognative overhead to those who don't know this mathematical structure while not adding any explanatory power. Furthermore for those who understand the structure you're possibly misleading them into thinking the choice of this structure is deep while preventing them from learning how to think about the field as everyone else does.
So longer then I initially expected (and missing points I was initially thinking of) but
tl dr: mathematics is unique in that in generally derives results from axioms rather than axioms from results like most other fields, this makes it very difficult to have concise terse texts that are foundational and totally rigourous (with exception of some theoretical physics and computer science).
These might require too much knowledge about the field, however maybe the mathematics in them might make it easier to read and be able to pull out the terms and facts that these experts seems to care about for future papers.
- Knot theory in understanding proteins
J. Math. Biol. (2012) 65:1187–1213
Actually targeted at mathematicians trying to get them to work in the field and has some references to biology which might help to just gain a bit of familarity/knowledge of what's important before reading other texts.
- String-net condensation:
A physical mechanism for topological phases
PHYSICAL REVIEW B 71, 045110 2005
Probably not accessible without some knowledge of quantum physics, but argues about using the objects of fusion categories to describe phases of matter.
- CRICK, F., WATSON, J. Structure of Small Viruses. Nature 177, 473–475 (1956). https://doi.org/10.1038/177473a0
I think this was the paper that used group theory to explain the structure of a virus shell. There is a later paper that I think extended it to Kac-Moody algebras and quasi-crystals for an abnormal class of viruses but I can't quickly find it.