The speaker is saying 'The fishmonger, calling me a thief..". (It seems the line is adapted, but the speaker seems 兵十. Cf. the original.)
と in 泥棒だと言って is clearly 引用の「と」, and requires a 終止形 ending for the preceding verb (or words that conjugate).
[Edit] Maybe naruto is right - this is not a 引用, but still it requiring a 終止形 should be the same. I found a seemingly relevant article.
Compare:
- その女の子をかわいいと言って : saying that the girl is pretty
- その女の子をきれいだと言って : saying that the girl is beautiful
- その女の子を美少女だと言って : saying that the girl is a pretty girl
All the bold words are in 終止形.
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A simpler (no) explanation is that the acceptable particles are determined by the verb. If this sounds fair enough, just remember it as pattern:
- call N1 N2 = N1をN2と呼ぶ / N1{を,が}N2だと言う.
Concretely:
- {俺が泥棒だ, 俺を泥棒だ, ?俺は泥棒だ, ?俺を泥棒}と言って
are all possible with (?) being less natural, but
- 俺を泥棒と呼んで
is the only option.
In short, だ in the question exists because いう (tends to) requires it.
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FYI: ごん狐 is possibly one of the few non-fairy-tale stories that close to everybody knows in Japan (as long as s/he learned from standard textbooks in primary school).