Who identified the four basic types of animal tissue?

This is only a partial answer. A good source on the history of animal morphology is Russell's Form and Function, freely available on Project Gutenberg.

It seems that the crucial step from Bichat's 21 types to 4, not just numerically but also methodologically, was taken by Theodor Schwann, one of the founders of modern histology, in Mikroskopische Untersuchungen über die Uebereinstimmung in der Struktur und dem Wachstume der Tiere und Pflanzen (1839). However, he had 5 rather than 4 types, and they do not match the modern ones. Nervous and muscular tissues fall under the same type, and he took the OP suggestion to separate blood and lymph into a separate type, distinct from other connective tissues. Bichat's classification "was macroscopical and physiological; he relied upon texture and function in distinguishing them rather than upon microscopical structure". In contrast, Schwann came to his types based on studying the emergence of the tissues from the ovum, and used cellular structure as a consistent criterion. An intermediate figure between Bichat ans Schwann was von Baer (1828), who already emphasized the connection between differentiation of germ layers and emerging types of tissues, although, like Bichat, he used textural rather than cellular characteristics in his classification.

Here is Russell's retelling of Schwann's reasoning:

"Five classes of tissue can be distinguished, according to the extent and manner of the modifications which the cells composing them have undergone. There are first of all independent and isolated cells, such as the corpuscles of the blood and lymph, not forming a coherent tissue in the ordinary sense. Next there are the assemblages of cells lying in contiguity with one another, but not in any way fused; examples of this class are the epidermal tissues and the lens of the eye. In the third class come tissues the cells of which have fused by their walls, but whose cell-cavities are not in continuity, such as osseous tissue and cartilage. In the tissues of the fourth class, comprising the most highly specialised of all, not only are the cell-walls continuous but also the cell-cavities; to this class belong muscle, nerve and capillary vessels. A fifth class, of rather a special nature, includes the fibrous tissues of all kinds. This is the first classification of tissues upon a cellular basis, and it marks the foundation of a new histology which took the place of the "general anatomy" of Bichat...

Schwann found it easy to demonstrate the cellular nature of the tissues of his first three classes. With the other two classes he had more difficulty. Fibres of all kinds, he considered, arose by an elongation of cells, which afterwards split longitudinally into long strips, forming as the case might be white or elastic fibrous tissue. Muscle-fibres and nerve-fibres were formed in a totally different way, by coalescence of cells; each separate muscle-fibre and nerve-fibre was thus a compound cell. Capillaries, Schwann held, were formed by cells hollowed out like drain-pipes, and 179set end to end—a mistaken view soon corrected by Vogt (Embryologie des Salmones, p. 206, 1842)."

Further studies of blastoderm differentiation (so-called germ-layer theory), which led to better understanding of epithelial tissues and distinguishing nervous and muscular ones, were undertaken, among others, by Remak in Untersuchungen über die Entwickelung der Wirbelthiere (1850-5), and further by O. and R. Hertwig in Studien zur Blättertheorie (1879-80).