To conclude, it is a major problem for a Muslim to claim to be a vegan or vegetarian – both practically and ideologically. If a Muslim wants to treat animals well, then in fact, they are following the Sunnah of the Prophet Muhammad ﷺ. If a Muslim would like to abstain from eating meat, it is entirely permissible according to Islam, even though they will miss out on attaining many rewards. However, to hold fast to all of the ideological concepts of veganism and vegetarianism, (namely the slaughter practice itself), will prevent a Muslim from performing required acts of worship, which is contradictory to our religion, and unacceptable before Allah.

The Prophet (ﷺ) is the best example of how a human being can be, and no other way is better, Alhamdulillah. We must let Allah’s word guide our passionate causes. We must not be tricked by shaytaan into thinking that slaughtering animals is backward or barbaric. Islam does not promote the ill-treatment of animals, but it also neither promotes abstinence from slaughter, veganism, nor vegetarianism. If a slaughter is implemented in the way of our Prophet (ﷺ) then it IS the definition of moral, humane, and animal rights.

Did our Prophet (ﷺ) slaughter animals? Yes. Does Islam promote unnecessary killing and harm of animals? No. Was our Prophet (ﷺ) kind and humane to animals? Yes. This is the best personality and the best way to encompass true love and respect for all beings. As Muslims, we must follow him and his way (ﷺ).
“Say, “If you ˹sincerely˺ love Allah, then follow me (the Prophet and his way); Allah will love you and forgive your sins. For Allah is All-Forgiving, Most Merciful.” [3:31]
The reality is, a ‘Muslim’ is one who submits his will to the will of Allah. Morality is not something that we determine it to be on our own. Rather, what is moral is that which Allah deems to be so. We submit to him not only in action, but in thought and concern as well. There will always be “attractive” new ideologies that pop up into the public space. Muslims must do their best to hold fast to what is in accordance with our religion, and embrace those aspects, and return those that contradict back to where they belong.

So, to answer the question which was asked in the beginning, “Can a Muslim be a Vegan or Vegetarian?”
Here is the final verdict:

Yes, you can be a Muslim without eating these animal products, but you need to be aware of the following:
You should not think that these things are haraam (forbidden), because Allah says:

يَـٰٓأَيُّهَا ٱلَّذِينَ ءَامَنُوا۟ لَا تُحَرِّمُوا۟ طَيِّبَـٰتِ مَآ أَحَلَّ ٱللَّهُ لَكُمْ وَلَا تَعْتَدُوٓا۟ ۚ إِنَّ ٱللَّهَ لَا يُحِبُّ ٱلْمُعْتَدِينَ
O you who believe! Make not unlawful the tayyibaat (all that is good as regards foods, things, deeds, beliefs, persons, etc.) which Allah has made lawful to you, and transgress not. Verily, Allah does not like the transgressors. [al-Maaidah 5: 87]

You should not think that it is better to abstain from these foods, or that doing so will be rewarded, or that a vegetarian is closer to Allah than others, and so on. It is not permitted to draw closer to Allah in this way. The Prophet ﷺ, who is the best of mankind and the closest to Allah, used to eat meat and drink milk and honey. When one of his Companions wanted to give up meat, he told him that this was wrong. Anas RA reported that there was a group of the Companions of the Prophet ﷺ, one of whom said, I will never marry women; another said, I will not eat meat; another said, I will not sleep on a bed; and another said, I will fast and never break my fast. News of this reached the Messenger of Allah ﷺ. He praised and thanked Allah, then said: What is wrong with the people who are saying such and such? I pray and I sleep; I fast and I break my fast; and I marry women. Whoever deviates from my Sunnah (way) does not belong to me. (Reported by al-Nisai; the story is also to be found in the two Sahehs of al-Bukhari and Muslim).

There is a great difference between not eating a certain kind of food because one does not like it, or it does not agree with one, or one has been put off for example by seeing an animal slaughtered when one was a child, which may leave one with a distaste for meat and other similar reasons, and thinking that meat is haraam and that abstaining from it is an act of worship, as is done by some groups.

Once this matter is clear in your mind, there is nothing wrong with not eating foods that you do not like.
And Allah knows best.