The earliest use of the phrase in this sense that I can find is from 1873, in a school textbook by William Smith and Theophilus Hall:
Shakspearian Sonnet.—In its less proper form the Sonnet is simply a poem of fourteen Heroic† lines, rhymed alternately and ending with a Couplet. The Sonnets of Shakspeare belong to this class.
William Smith and Theophilus D. Hall (1873). A School Manual of English Grammar, p. 168. London: John Murray.
† “Heroic” in the sense “traditionally used in poetry with mythological or historical heroes as its subject matter” (OED), so that a “Heroic line” means “iambic pentameter” in the context of English poetry.
Textbooks normally collect common knowledge rather than inventing new terms, suggesting that the phrase ought to have been in circulation with this meaning prior to 1873. But if it was, I have not been able to find any examples. I did find one near miss:
I am aware that many strict critics object to the term “Sonnet” being applied to any poems but those written in the exact Italian measure; unjustly, as I think, since the rich cadences and extreme facility of rhyming which make the Petrarchian stanza so easy and so beautiful in the original, do not exist in English. […] Milton,—whose taste was formed in the Tuscan schools, adhered closely to the Italian model. But Shakspeare—master of no melody but that of his native tongue, in which he reigned, and still reigns, without a rival—has left us upwards of one hundred and fifty “Sonnets,” in the simple measure of three alternate quatrains, closed by two heroic lines; evidently considering the title generally applicable to all poems which follow Petrarch so far as to consist of a single thought carried through fourteen lines, however their rhythm may be modified to suit the necessities of our language. I am inclined to think the Shakspearian stanza a better English model than that adopted by Milton.
Caroline Sheridan Norton (1841). The Dream, and Other Poems, p. 281. London: William Clowes.
It took a few years after Smith and Hall’s textbook for the term to catch on. I’ll give a selection of early uses.
After this jubilant little fragment came a sonnet, “Written before re-reading King Lear,” and a Shaksperean sonnet (“When I have fears that I may cease to be”) which reads to me like an exercise of fancy.
R. H. Stoddard (1878). ‘After Many Days: A Study of Keats’. In Scribner’s Monthly, January 1878, p. 409.
In an obituary notice† of the late Charles Turner—Alfred Tennyson’s elder brother—I find the following description of his character as a poet. ‘He had a very considerable gift of tender fancy and of plaintive elegiac melody, but he was lacking in a sense of style; his writings are chiefly sonnets, and they are mostly very incorrect in form.’
As to his gift of fancy and melody, everybody who has read any of his poems will agree. But in saying that his sonnets are ‘incorrect in form’ the writer can only have meant that the rhymes do not follow the order prescribed either by the Miltonic or the Shakespearian sonnet; and in imputing to him a lack of the ‘sense of style’ he must, I suppose, have meant the same thing; for otherwise it would be hard to say what he can have meant.
James Spedding (1879). ‘Charles Tennyson Turner’. In The Nineteenth Century, September 1879, p. 461.
† The Academy, 10 May 1879, pp. 412–413.
Spedding in the quote above uses the phrase “Shakespearian sonnet” in the sense “sonnet employing the same rhyme scheme as Shakespeare” without feeling the need to gloss it. But this sense of the phrase must not have been commonplace in 1879, for one of Spedding’s readers was sufficiently puzzled to write a sharp response questioning whether the phrase is appropriate at all:
Mr. Spedding’s notice of “Charles Tennyson Turner” […] contains some oddities. Mr. Spedding is very angry with the criticism on the late Charles Turner, that his sonnets are “incorrect in form,” and that he was “lacking in the sense of style.” He thinks this can only mean that the sonnets do not “follow the order prescribed either by the Miltonic or the Shaksperian sonnet.” Does Mr. Spedding imagine that Milton and Shakspere invented the sonnet? We should have supposed that in adopting a decidedly artificial system of verse it might be thought not altogether superfluous to follow its form exactly. A quatorzain of any arrangement may be an admirable poem, no doubt; that no one denies. But Gautier’s remark, “Write sonnets properly, or don’t write them at all,” remains applicable.
Anon (1879). ‘Magazines and Reviews’. In The Academy, 6 September 1879, p. 176.
Some of the sonnets are very, I must say unpardonably, licentious in form. I recognise stricter and looser forms and the Shakespearian sonnet, though it is a sonnet only in genre and not one if by sonnet you mean the Italian sonnet, which is the sonnet proper—but this is a question of names only—the Shakespearian sonnet is a very beautiful and effective species of composition in the kind. But then, though simpler, it is as strict, regular, and specific as the sonnet proper. Moreover it has the division into the two parts 8+6, at all events 4+4+4+2. Now it seems to me that this division is the real characteristic of the sonnet and that what is not so marked off and moreover has not the octet again divided into quatrains is not to be called a sonnet at all.
Gerard Manley Hopkins (12 October 1881). Letter to Richard W. Dixon. In The Correspondence of Gerard Manley Hopkins and Richard W. Dixon, p. 71. Oxford University Press.
Contemporaneously with Wyat, the Earl of Surrey produced poems constructed upon the model subsequently known as Shakspearean; and, following Surrey, Spenser wrote a series similar in scheme and scope, but with a variation in the arrangement of rhymes.
Hall Caine (1882). Sonnets of Three Centuries, p. x. London: Elliot Stock.
What is styled the Shakespearian sonnet is so called only out of deference to the great poet who made such noble use of it: in the same way as Petrarca is accredited with the structural form bearing his name. As “the sweete laureate of Italie”† had predecessors in Guittone d’Arezzo and Amalricchi, so Shakespeare found that the English sonnet—as it should be called—having been inefficiently handled by Surrey, discarded by Spenser, taken up and beautified by Sir Philip Sydney (who seemed unable to definitely decide as to what form to adopt), was at last made thoroughly ready for his use by Daniel and Drayton.
William Sharp (1886). Sonnets of this Century, p. xlv. London: Walter Scott.
† Petrarch, as described in the ‘Prologue to the Clerk’s Tale’ in Geoffrey Chaucer’s Canterbury Tales: “Francis Petrarc, the laureate poet, / Highte this clerk, whose rhetoric so sweet was called / Illumin’d all Itale of poetry”.