Would the husband be sinful if he did not prevent his wife from slaughtering for other than Allah, and would he in this case be considered a polytheist like her?

Slaughtering for other than Allah is polytheism (shirk) because it is an act of worship that is not permissible to direct to anyone but Him, as evidenced by the verses: ﴿Say, "Indeed, my prayer, my rites of sacrifice, my living and my dying are for Allah, Lord of the worlds. No partner has He. And this I have been commanded, and I am the first of the Muslims."﴾ (Quran 6:162-163) and ﴿So pray to your Lord and sacrifice [to Him alone].﴾ (Quran 108:2), and the hadith: "Allah curses whoever slaughters for other than Allah." If the intention is to glorify someone other than Allah for whom the slaughter is made, then it is disbelief (kufr) and apostasy. Therefore, you must prevent your wife from this polytheistic and disbelief-laden act, guide her, advise her, forbid her from this evil, explain the danger of her action, and supplicate for her guidance.