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!
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.
Saying السلام عليكم أيها النبي in tashahud?
The (plural) version you cited السلام عليكم أيها النبي is unknown to me and as you mentioned it sounds wrong and even a search on Sunna.com shows that this version is not quoted in any (major) hadith collection.
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).
I found the plural form on this site and in the Fatwa 531 of Sheikh ibn 'Othaymyn on this site (both without quoting a source), but some scholars point at the third possible explanation for the use which i will explain later!
About your shirk matter:
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!
Now about the matter you addressed السلام عليكم seems wrong as we are addressing our Messenger (May peace be upon him) as a single person! I think i have to possible explanations:
- The plural form is a kind of way to honor him as in Arabic the equivalent a way of the honorific speech is using a plural انتم (examples: سيادتكم, فخامتكم) so you use this might be a reason for that.
- An other reason might be as we do the salat 'ala an-Nabiy الصلاة على النبي we often include his family (Ahl-al-Bait) or even his sahaba in it so a plural form would be acceptable.
- A third reason -which is related to my first point- is that the plural form السلام عليكم is considered and used for honoring the death based on this riwaya of the hadith (see also in Suann abi Dawod and Sunan ibn Majah) where we are taught what to say when visiting the graveyard!
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!
Here's one fatwa where the matter of discussion is whether the expression السلام عليك أيها النبي or السلام على النبي and if any is related to shirk! And here is one about using an expression for greeting which differs from what our Messenger has taught us!
And Allah knows best.