Leaving aside the "with Chinese characteristics" part, China is nominally a socialist-wanting-to-become-communist country.
Pretty much all the former Soviet bloc countries defined themselves as democratic. Their remnants, even those with [some] special characteristics, like North Korea (DPRK) and Cuba, still define themselves as democratic. To quote Cuba's 1976/2002 constitution for instance:
Cuba is a socialist State of workers, independent and sovereign, organized with all and for the good of all, as a united, democratic republic, for the enjoyment [...]
The wording in their 2019 constitution has been changed somewhat to even emphasize the democratic part earlier on:
Cuba is a democratic, independent and sovereign socialist State of law and social justice, organized by all and for the good of all [...]
So, on the level of an extremely broad/nominal definition of democracy, China is nothing special, despite your assertion "It seems odd that China consider itself to be democratic". If that's what the question is. If the Q is what country or countries is closest in system with China and why the Western think tanks typically rank it at the bottom of their democracy scale[s], that's a slightly different question.
If you drill down to article 5 or so of Cuba's constitution, the "democratic character" is clarified to be in the vanguard party. China gets some bonus points for clarity here, as it's mostly spelled out in the first article of their constitution:
The People's Republic of China is a socialist state governed by a people's democratic dictatorship that is led by the working class and based on an alliance of workers and peasants.
And while "democratic dictatorship" is probably not a common term elsewhere, article 3 of China's constitution mentions democratic centralism, together with elections. What this meant in practice in most such one-party countries was that elected officials were typically limited to a pool mostly agreeing with the leadership (du jour) of the party. Although factions did (in fact often) emerge in communist parties, typically it was around personalities or minor divergences in Marxist-Leninist doctrine, although China is a bit more special in this regard, with their economic doctrine having diverged to a substantial degree and for a long time.