If you look at books on (not just Abelian) harmonic analysis, such as Hewitt and Ross, they tell that Tannaka-Krein duality was originally a non-Abelian version for compact topological groups of Pontryagin duality for locally compact Abelian topological groups. The Pontryagin dual is really a group (it consists of all continuous characters with pointwise multiplication), and allows one, for instance, to define a notion of Fourier transform on such groups. More generally, the whole topic of duality for locally compact topological groups is a blend of algebra and analysis, just as Schwartz's theory of distributions. Since Tannaka-Krein duality was first formulated in this way, this explains the (harmonic) analyst's interest on the topic.
It was noticed later that the same framework for Lie groups, formulated in the language of Hopf algebras (I believe this was done for the first time in Hochschild's book "The Structure of Lie Groups". Hewitt-Ross use the older terminology "Krein algebras"), could be extended to algebraic groups.