Yes, and no.
Answering the title and the first sentence specifically, the Shechina, Hashem's dwelling in the spiritual and physical worlds, can certainly be in multiple places at once.
Avot 3:6 demonstrates that Hashem will be present among every gathering of 10 Jews*, and even 5, 3 and 1 Jew(s), and brings pasukim to source these claims. Shemot 20:21 is the most striking:
בכל המקום אשר אזכיר את שמי אבא אליך וברכתיך
In every place where I cause My Name to be mentioned I will come unto
you and bless you
Which Sforno, Ohr HaChaim and others explain means that Hashem, when He promised to "dwell among us" (Shemot 25:8), initially meant dwell in more places than just between the Keruvim, such as within every Jew.
There is also the concept that Hashem's Presence sits among 3 judges when they judge. This is also brought in Avot 3:6, with a source in Tehillim.
That's the yes. The no is brought by Rabban Gamliel to a כופר in Sanhedrin 39a:
א"ל כופר לר"ג אמריתו כל בי עשרה שכינתא שריא כמה שכינתא איכא קרייה
לשמעיה מחא ביה באפתקא א"ל אמאי על שמשא בביתיה דכופר א"ל שמשא אכולי
עלמא ניחא ומה שמשא דחד מן אלף אלפי רבוא שמשי דקמי קודשא בריך הוא ניחא
לכולי עלמא שכינתא דקב"ה על אחת כמה וכמה
The emperor said to Rabban Gamliel: You say that the Divine Presence (Shechina)
dwells in any place where there are ten adult male Jews. He asked,
sarcastically: How many Shechinas are there? Rabban Gamliel
summoned the servant of the emperor and hit him on his neck
[be’appatka]. Rabban Gamliel said to him: Why did you allow the sun to
enter the house of the emperor? The emperor said to him: The sun rests
upon all the world; no one can prevent it from shining. Rabban Gamliel
said to him: And if the sun, which is one of ten thousand attendants
that are before the Holy One, Blessed be He, rests upon all the world,
the Divine Presence of the Holy One, Blessed be He, all the more so
rests upon the world. [Steinsaltz trans. and commentary]
So, it's not actually true that He's in multiple places at once, in the sense that He splits up into multiple parts, chas veshalom, but that does not preclude Him from shining the Light of His Presence, as it were, into wherever is zoche, and desirable to Him.
* See also Tanya Iggeret HaKodesh 23. Also brought in halacha, e.g. Shulchan Aruch HaRav. Related is that the Shechina rests on the hands of Cohanim during Birchat HaCohanim. There are many examples like this.