2 added 35 characters in body
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Charcoal, 2129 bytes

∧⁼§θ⁰§η⁰⁼↧⪫⪪θ_ω↧⪫⪪η_ω∧⁼§θ⁰§η⁰⁼↧⪫⪪⪫⪪θ_ω-ω↧⪫⪪⪫⪪η_ω-ω

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This prints a - for truthy and nothing for falsey.

Link to the verbose versionverbose version. It first compares the first character of both input strings (⁼§θ⁰§η⁰) and then compares the rest of both strings after removing the underscores and the hyphens (⪫⪪θ_ω⪫⪪⪫⪪θ_ω-ω) and converting to lowercase ().

Charcoal, 21 bytes

∧⁼§θ⁰§η⁰⁼↧⪫⪪θ_ω↧⪫⪪η_ω

Try it online!

This prints a - for truthy and nothing for falsey.

Link to the verbose version. It first compares the first character of both input strings (⁼§θ⁰§η⁰) and then compares the rest of both strings after removing the underscores (⪫⪪θ_ω) and converting to lowercase ().

Charcoal, 29 bytes

∧⁼§θ⁰§η⁰⁼↧⪫⪪⪫⪪θ_ω-ω↧⪫⪪⪫⪪η_ω-ω

Try it online!

This prints a - for truthy and nothing for falsey.

Link to the verbose version. It first compares the first character of both input strings (⁼§θ⁰§η⁰) and then compares the rest of both strings after removing the underscores and the hyphens (⪫⪪⪫⪪θ_ω-ω) and converting to lowercase ().

1
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Charcoal, 21 bytes

∧⁼§θ⁰§η⁰⁼↧⪫⪪θ_ω↧⪫⪪η_ω

Try it online!

This prints a - for truthy and nothing for falsey.

Link to the verbose version. It first compares the first character of both input strings (⁼§θ⁰§η⁰) and then compares the rest of both strings after removing the underscores (⪫⪪θ_ω) and converting to lowercase ().