I read in the siddur it says to love Hashem’s name or to trust in Hashem’s name. Why His name and not Hashem himself?
Loving Him and loving His Name go hand in hand. As I was reading in a passage by the Riva in shul (I believe it was in a biography about Abir Yaacov), Hashem's Name is revealed, but His Essence is concealed i.e. He Himself is concealed, but we know His Name. We know Him by His Name. Does that make sense? (He went on to say, interestingly, that we are the same: we have a soul and a body, and the body is revealed but the soul is concealed, yet through the body we can know the soul 🤯)
Saying we love His Name isn't any different to saying we love Him. Same is true for trust.
I don't know what you saw to base this on, but here's a good example from Tehillim:
וְאֹהֲבֵ֥י שְׁ֝מ֗וֹ יִשְׁכְּנוּ־בָֽהּ׃
lovers of His Name shall dwell there [in Zion].
Kav HaYashar writes, after discussing the humility of Rabbi Yehudah and Rabbi Yitzchak for listening to the Torah of a child (Rav Hamnuna the Elder's son):
For a man who renders himself like the dust upon which all trample is called, “Beloved above and valued below.” Moreover, he is called one of the “lovers of Hashem,” concerning whom it is written, “And the lovers of His Name will dwell in it [in Tzion]” (Tehillim 69:37).
Lovers of Hashem = Lovers of His Name.
The pasuk we are quoting is talking about His Name that is Elokim:
כִּ֤י אֱלֹקִ֙ים ׀ י֘וֹשִׁ֤יעַ צִיּ֗וֹן וְ֭יִבְנֶה עָרֵ֣י יְהוּדָ֑ה וְיָ֥שְׁבוּ שָׁ֝֗ם וִירֵשֽׁוּהָ׃
For God will deliver Zion and rebuild the cities of Judah; they shall live there and inherit it;
And in the same chapter:
אֱֽלֹקִ֗ים אַתָּ֣ה יָ֭דַעְתָּ לְאִוַּלְתִּ֑י וְ֝אַשְׁמוֹתַ֗י מִמְּךָ֥ לֹא־נִכְחָֽדוּ׃
God, You know my folly; my guilty deeds are not hidden from You.
...we say "You" (i.e. Him, Himself)
When His Name is One, so will He be One as Zacharia states:
בַּיּ֣וֹם הַה֗וּא ה' יְהֹוָ֛ה אֶחָ֖ד וּשְׁמ֥וֹ אֶחָֽד
and of course His Names are One as it is written:
כִּ֤י ה' ה֣וּא הָֽאֱלֹקִ֔ים
See this amazing ma'amer for more information about this Oneness, taking it up to Him, Himself.
Tanya writes on the first quoted pasuk about those who love Hashem's name:
Hence he does not need to concentrate so much on the greatness of G–d to consciously create a spirit of knowledge and fear of G–d in his mind in order to guard himself against violation of the prohibitive commandments or to arouse the love of G–d in his heart to induce his attachment to Him through the fulfillment of the [positive] commandments and the study of the Torah, which balances everything else. For him suffices the hidden love that is in the heart of all Jews, who are called “those who love His name.”
The mussar and chassidus passages quoted here are very interesting, and they both demonstrate that "lovers of His Name" is an idiom to refer to people who have an exceptional quality in their love of Him.
One last point. The idea of a Name in Torah isn't just a pointer to a person. It also represents a person's mission, purpose, and what they are all about. This topic comes up a lot but I have rarely heard it so well explained than this shiur by Rabbi Tovia Singer, discussing one of Hashem's Names: Muslim Asks Rabbi Tovia Singer: Why Did Moses Ask God for His Name? Didn’t the Jews Know God’s Name?. I assume this will be relevant to your understand of what it means to love Hashem's Name - it's even more than just loving Him, it's loving His Plan, His Purpose, His Way - everything about Him!