Once upon a time I imagine people were very happy to think of Lie algebras as “infinitesimal groups,” but presumably when infinitesimals fell out of favor this interpretation did too. In this post I’ll record an observation that can justify thinking of Lie algebras as groups in a strong sense: they are group objects in a certain category which can be interpreted as a category of “infinitesimal spaces.”
Below we work throughout over a field of characteristic zero.
For starters, the universal enveloping algebra functor 
where 
The key observation is that this upgraded functor 
Hence Lie algebras embed as a full subcategory of Hopf algebras; that is, they can be thought of as Hopf algebras satisfying certain properties, rather than having extra structure (in the nLab sense). What are these properties? For starters, they are all cocommutative. This is important because cocommutative Hopf algebras are group objects in the category of cocommutative coalgebras (this is not true with “cocommutative” dropped!), which in turn can be interpreted as a category of infinitesimal spaces. (For example, this category is cartesian closed, and in particular distributive.)
Hence Lie algebras are group objects in cocommutative coalgebras satisfying some property (for example, “conilpotence”; see Theorem 3.8.1 here).
Annoying Precision 
What do you mean infinitesimals went out of fashion? Artin’s Algebra textbook gives a totally rigorous construction of the Lie algebra of a linear algebraic group by looking at the groups G(k), G(k[e]/e^2), and $G(k[e]/e^3). I imagine it’s pretty close to what Lie originally did. (Artin plays a shell game about the difference between a Lie group and a linear algebraic group.)
Not to ruin the joke or belabor the obvious, but if you take ind-finite cocommutative coalgebras and take the linear dual, you get pro-finite commutative algebras (with the completed tensor product) and their spectra are called formal schemes and group objects in them are called formal groups and are a pretty popular way of making sense of “infinitesimal groups.” (You might want to link between your posts on distributions, but I don’t think you get to the end of the story.)
Not super relevant, but while I’m on the topic: the universal enveloping algebra is the ring of left-invariant differential operators on the group and its dual, the ring of functions on the formal group, is the formal completion at the identity of functions on the group. The pairing is to apply the operator to the function to get a new function and evaluate at the identity.
Nice post! Could you say more about more the phrase: ”… which in turn can be interpreted as a category of infinitesimal spaces” by the end of the post?
Over a field, the category of cocommutative coalgebras is the ind-completion of the category of finite-dimensional cocommutative coalgebras, which is equivalent to the opposite of the category of finite-dimensional commutative algebras.