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السلام عليكم is often translated as peace and Allah's mercy and blessing be upon you. Scholars(here an Explanation of the words of tashahud by ibn Baz ) say that in السلام عليك أيها النبي, salam refers to السلامة which may mean safety, well being etc.: so by this we ask Allah to protect and keep his His Messenger (peace be upon him) safe from any evil and have mercy on him. This is what i could add to the already given answers on the meaning as you already noticed the major difference plural vs singular!

السلام عليكم is often translated as peace and Allah's mercy and blessing be upon you. Scholars  (here an Explanation of the words of tashahud by ibn Baz ) say that in this special context in السلام عليك أيها النبي, salam refers to السلامة which may mean safety, well being etc.: so by this we ask Allah to protect and keep His Messenger (peace be upon him) safe from any evil. This is what i could add to the already given answers on the meaning as you already noticed the major difference is plural vs singular!

And finally in this case shirk isn't a matter as yes we believe in one God, but it's not him who is addressed by these words. On the other Hand assuming the plural form is correct i think i have given evidences that it could be considered either to honor or to add Ahl-al-Bait (May Allah be pleased with them) into the tashahud which is totally acceptable because of the salat al ibrahimya (for example, which is a part of the tashahud and addresses both Aal-Ibrahim and Aal-Mohammad (May peace be upon both of them)): so shirk seems out of question, unless your intention was something different!

And finally in this case shirk isn't a matter as yes we believe in one God, but it's not him who is addressed by these words, as we rather ask him to have mercy and protect our beloved Prophet (Peace be upon him). On the other Hand assuming the plural form is correct i think i have given evidences that it could be considered either to honor or to add Ahl-al-Bait (May Allah be pleased with them) into the tashahud which is totally acceptable because of the salat al ibrahimya (for example, which is a part of the tashahud and addresses both Aal-Ibrahim and Aal-Mohammad (May peace be upon both of them)): so shirk seems out of question, unless your intention was something different!

I think you will find the meaning of the words or even have found it out already, so i will not explain it, and start analyzing the two opposite sentences السلام عليكم أيها النبي and السلام عليك أيها النبي from a fiqh and a linguistic POV.

Maybe it's a printing error or a special case for a special madhab as the well known and good referenced version is the singular form السلام عليك أيها النبي you may find evidences for an other version السلام على النبي for example in Muwatta' Malik. These both versions are acceptable both in Arabic language and at situation of the tashahud in prayer (See also my linked fatwas).

We know that often we use the plural form السلام عليكم even if we greet one person this is also according to the rules of the Arabic language acceptable, as in cases the singular can be replaced by the plural and vice versa and we have many examples of this in the Quran you may find it out if you read the Quran in different riwayat for example in Surat al-Baqara, al-A'raaf, Ibrahim, al-Hijr, al-Kahf, al-Furqan, an-Naml, a-Rum, Fatir, ash-Shura and al-Jathiyah some riwaya it's the wind/storm= الريح in others the winds = الرياح and even in surat al-Kahf (18) in the story of the two man among one was apparently a believer and the other a rich man who had two gardens جنتين some riwayat seem to forget that the man had two of them and go ahead considering one garden جنته others go ahead speaking in the dual plural ( لأجدن خيرا منها منقلبا kufi scripture vs. ** منهما** Makky, Madani and Shami scripture) form!

Meaning

السلام عليكم is often translated as peace and Allah's mercy and blessing be upon you. Scholars(here an Explanation of the words of tashahud by ibn Baz ) say that in السلام عليك أيها النبي, salam refers to السلامة which may mean safety, well being etc.: so by this we ask Allah to protect and keep his His Messenger (peace be upon him) safe from any evil and have mercy on him. This is what i could add to the already given answers on the meaning as you already noticed the major difference plural vs singular!

Now I will start analyzing the two opposite sentences السلام عليكم أيها النبي and السلام عليك أيها النبي from a fiqh and a linguistic POV.

The well known and good referenced version is the singular form السلام عليك أيها النبي you may find evidences for an other version السلام على النبي for example in Muwatta' Malik. These both versions are acceptable both in Arabic language and at situation of the tashahud in prayer (See also my linked fatwas).

We know that often we use the plural form السلام عليكم even if we greet one person this is also according to the rules of the Arabic language acceptable, as in cases the singular can be replaced by the plural and vice versa and we have many examples of this in the Quran you may find it out if you read the Quran in different riwayat for example in Surat al-Baqara, al-A'raaf, Ibrahim, al-Hijr, al-Kahf, al-Furqan, an-Naml, a-Rum, Fatir, ash-Shura and al-Jathiyah some riwaya it's the wind/storm= الريح in others the winds = الرياح and even in surat al-Kahf (18) in the story of the two man among one was apparently a believer and the other a rich man who had two gardens جنتين some riwayat seem to forget that the man had two of them and go ahead considering one garden جنته others go ahead speaking in the dual plural ( لأجدن خيرا منها منقلبا kufi scripture vs. ** منهما** Makky, Madani and Shami scripture) form!

Using plural instead of singular and vice versa is common in Arabic language!