Jelly, 14 bytes

ŒgŒQ¬TịƲLÐṀÆṃ'

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How it works?

ŒgŒQ¬TịƲLÐṀÆṃ' – Full program. Receives a list of digits as input.
Œg             – Group equal adjacent values.
  ŒQ           – Distinct sieve. Replace the first occurrences of each value by 1
                 and the rest by 0. [1,2,3,2,3,2,5]ŒQ -> [1,1,1,0,0,0,1]       
    ¬T         – Negate and find the truthy indices.
      ịƲ       – Then index in the initial list of groups.
               – This discards the groups that only occur once.
        LÐṀ    – Find all those which are maximal by length.
           Æṃ' – And take the mode.

Jelly, 12 bytes

Œgœ-Q$LÐṀÆṃ'

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Previous version – 14 bytes

ŒgŒQ¬TịƲLÐṀÆṃ'

Try it online!

How it works?

Œgœ-Q$LÐṀÆṃ' – Full program. Receives a list of digits as input.
Œg           – Group equal adjacent values.
  œ-Q$       – Multiset difference with itself deduplicate.
      LÐṀ    – Keep those that are maximal by length.
         Æṃ' – Mode. Returns the most common element(s).
-------------------------------------------------------------------------
ŒgŒQ¬TịƲLÐṀÆṃ' – Full program. Receives a list of digits as input.
Œg             – Group equal adjacent values.
  ŒQ           – Distinct sieve. Replace the first occurrences of each value by 1.
                 and the rest by 0. [1,2,3,2,3,2,5]ŒQ -> [1,1,1,0,0,0,1]       
    ¬T         – Negate and find the truthy indices.
      ịƲ       – Then index in the initial list of groups.
               – This discards the groups that only occur once.
        LÐṀ    – Find all those which are maximal by length.
           Æṃ' – And take the mode.

Jelly, 14 bytes

ŒgŒQ¬TịƲLÐṀÆṃ'

Try it online!

How it works?

ŒgŒQ¬TịƲLÐṀÆṃ' – Full program. Receives a list of digits as input.
Œg             – Group equal adjacent values.
  ŒQ           – Distinct sieve. Replace the first occurrences of each value by 1
                 and the rest by 0. [1,2,3,2,3,2,5]ŒQ -> [1,1,1,0,0,0,1]       

Jelly, 14 bytes

ŒgŒQ¬TịƲLÐṀÆṃ'

Try it online!

How it works?

ŒgŒQ¬TịƲLÐṀÆṃ' – Full program. Receives a list of digits as input.
Œg             – Group equal adjacent values.
  ŒQ           – Distinct sieve. Replace the first occurrences of each value by 1
                 and the rest by 0. [1,2,3,2,3,2,5]ŒQ -> [1,1,1,0,0,0,1]       
    ¬T         – Negate and find the truthy indices.
      ịƲ       – Then index in the initial list of groups.
               – This discards the groups that only occur once.
        LÐṀ    – Find all those which are maximal by length.
           Æṃ' – And take the mode.