How can Eph. 5:31-32 say that marriage refers to Christ and the Church without saying it was instituted to signify that?
Here is an excerpt from Turretin on why marriage is not a sacrament, perhaps it may shed some light on your own question:
“XL. The fifth sacrament is matrimony, according to the Council of Trent (Session 24, Canon 1, Schroeder, p. 181). But here all the requisites of a sacrament are equally wanting.
(1) The divine institution, since it is evident that from the first creation it was ordained for the propagation of the human race (from which end it has not turned in the New Testament, Mt. 19:4, 5). (2) The element, which may be a sacramental sign ordained by Christ. (3) The word of promise of grace, peculiarly annexed to marriage, that it may be confirmed and applied by it. (4) The ordinary and perpetual use in the New Testament church alone, while marriage according to the law of nature and the Gentiles is common both to believers and unbelievers and belongs to both churches.
XLIII. The passage of Paul in Eph. 5:32 is falsely wrested to human marriage. He expressly interprets this himself concerning the mystical union of Christ with his church, which both the form of exclamation (having a feeling [pathos] of admiration) and the whole preceding context from vv. 26, 27 evince. The word “mystery” is improperly translated sacrament because that word usually declares in the Scriptures not a sign, but a spiritual thing sealed. And it extends more widely than sacrament strictly so called, for not every mystery can be called a sacrament, but only that which offers the grace of salvation by a word of promise and confirms it by an external symbol divinely instituted.
XLIV. Whatever represents a sacred thing is not immediately a sacrament. For thus the sun and other things of the same kind representing Christ would be sacraments. Nor is whatever has an annexed promise of any grace a sacrament; otherwise almsgiving and prayer would be sacraments. Finally, the promise of grace connected with marriage is temporal concerning the raising up of seed, not sacramental. ”
Excerpt From
Institutes of Elenctic Theology (Vol 3)
Francis Turretin
A further quote may be helpful from Flavel's "Husbandry Spiritualised":
"As man is compounded of a fleshly and spiritual substance, so God hath endowed the creatures with a spiritual, as well as fleshly usefulness; they have not only a natural use in Alimental and Physical respects, but also a spiritual use, as they bear the figures and similitudes of many sublime and heavenly mysteries."
Source:
https://quod.lib.umich.edu/e/eebo/A39665.0001.001/1:2?rgn=div1;view=fulltext