The lyric says:
I wonder what オレ means. Is it a different way to write: "俺"?
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オレ, おれ and 俺 are used by males and mean “I” or “me”. Their meanings are same but their nuances are different.
Whether オレ, おれ or 俺 is used depends on the speaker’s or the author’s preference.
The lyricist of the song could have written it in other ways like so:
But he chose オレはこいつと, maybe because it fits the character's personality and the song.
A character using オレ is generally a boy, a young man or an internally young man. Sometimes オレ expresses the boy’s active personality, or sometimes オレ expresses the boy’s casualness (e.g. The boy’s attitude is always somewhat casual, almost never completely formal.)
The characteristics of characters using 俺 or おれ are diverse.
A character using 俺 is generally older or more intelligent or more something than a character using オレ but there are many exceptions. A character related to Japanese traditional culture tends to use 俺 rather than オレ.
Many authors tend to use オレ rather than おれ to write a character’s line when 俺 is not an option, because オレ is katakana so it makes distinguishing the word from other hiragana words easier. For example,
「さっきおれのこと呼んだ？」(Did you call me a while ago?)
さっきオレのこと is a bit easier to read. But it wears the word オレ’s nuance, so if the nuance doesn’t fit the character's personality or situation, the author won’t use オレ in the character’s line.
Some authors have their special policy to use おれ, オレ or 俺. In that case, the word おれ, オレ, or 俺 may express the special nuance.
For example, a lot of male characters in a popular manga One Piece use おれ. The author of the manga, Eiichiro Oda (尾田栄一郎), answered a fan’s question why he uses おれ rather than 俺 or オレ. What he said is that hiragana おれ makes him feel a belief. The following is his exact words.
（『ONE PIECE 巻五十二』第503話後の質問コーナーSBSより抜粋）
Therefore, each of オレ, おれ, and 俺 has its general nuance expressing the speaker’s characteristics, but in some cases it can also have a special nuance which the author or the speaker intends, although they have the same meaning “I” or “me”.
It is as you think but while it may be a different way to write 俺 it is orthographically non-standard. Let's also the following facts in the ポケットモンスター case:
"オレは" = I (nominative case)
"オレ" = me (objective case)
"俺" is pronounced "オレ" in Katakana, Japanese.