The Mishnah says "האיש מקדש", just like on Shabbos. Nevertheless, on Shabbos we call it a Kiddush (single) but on wedding we call it either Kiddushin, Erusin, Nissuin which are all plural.
R. Yissakhar Tamar addresses the term kiddushin in Alei Tamar to Yerushalmi Berakhot (2:3), and his explanation answers erusin as well.
He explains that the plural expression references the three ways that kiddushin can be performed (cf. Kiddushin 1:1). However, he notes that Teimanim traditionally referred to it in the singular.[i]
It should be noted that this explanation probably does not address nissuin though.
[i] Lest anyone ask that the tractate is called 'Kiddushin', that is like 'Pesahim', the korban is still called a 'Pesah'. So too, the tractate refers to marriages, but individual ones are referred to as 'kiddush'.
Kiddush means Havdolo(separation).Rashi says in Bereishis 2,3 on the word וַיְקַדֵּשׁ -קדשו במן שלא ירד כלל בשבת - Hashem separated Shabbos from the rest of the week by not making the manna fall at all on shabbos in contrast to the rest of the week where it did fall.
On shabos You are doing one separation between a Chol weekday, and a Kodesh Shabbos this is indicated in the Brocho where we say Mekadesh Hashabbos (note that Ki vonu bocharto.. micol hoamim is in brackets and some people don't say it for this very reason that we are mainly marking the difference between Shabbos and Chol, other separations are subordinate. This would also explain why in havdala [singular] that even though there are 4 separations, the preeminent separation is between Shabbat and Chol).
This is 1 division
At a kidushin there are numerous divisions, because through marrying a women That man becomes separted (see Vayikra chapter 18 for the list)i.e biblicly forbiden to:
1. her mother,
2.her Mother's mother
3.her Father's Mother
5.her Son's daughter(s)
6.her daughters daughter(s)
7. Her sister(s)
8. He's Rabbinically forbidden to the Bride until she enters his domicile (i.e chuppa).
She is also forbiddon to everyone in the world like Hekdesh.
So we see 9 different separations: 8 on the mans side and 1 separation to an entity(which includes all men including the husband who is rabbically forbidden to her until she enters his domicile) on the womans side.
This is indicated in the Brocho (kesubos 7b) ברכת האירוסין מאי מברך רבין בר רב אדא ורבה בר רב אדא תרוייהו משמיה דרב יהודה אמרי בא"י אמ"ה אשר קדשנו במצותיו וצונו על העריות ואסר לנו את הארוסות והתיר לנו את הנשואות על ידי חופה וקדושין - Arayos are new forbidddon relationships that are biblically forbiddon, arusos are the man and woman who are rabbically forbiddon to each other until she enters his domicile.
"האיש מקדש" is singular because he is only one man (we're not dealing with many men betrothing simultaniously). But he can sanctify (separate) many things at once e.g האיש מקדש את אשתו ואת עצמו מן אמה אם אביה בתה בת בתה
Eirusin is plural because it means 2 things are happening the husband is betrothed to the woman and the woman is betrothed to the man, result being they are both ossur to each other as it says ואסר לנו את הארוסות.
Nissuin is plural because it means 2 things are happening the husband is permitted (i.e the prohibition is lifted of himself) to the woman and the woman is permitted to the man, result being they are both permitted to each other. Nissuin litterally means lifting off oneself (the prohibition of being together from the time of betrothal) as it says famously in Tashlich from micha 7,19 מִי אֵל כָּמוֹךָ נֹשֵׂא עָוֹן - Who is G-d like You who lifts off sin from upon us.