There are two types of wallets:
A full node client uses the peer-to-peer network to validate transactions and blocks and to relay information to other peers. To verify all transactions, the entire blockchain (~145G) acts as a dependency to enable the tracing of transactions on the network.
SPV, Simplified Payment Verification, is used as a light-weight implementation of Bitcoin only validating your transactions by asking for data from full node peers. We use SPV clients for convenience to prevent accumulating about 145G when validating only one address worth of transactions.
Does anyone need to download the full blockchain to use Bitcoin on a mobile phone?
No. The wallet software you are talking about is a Full Node and is, by design, not available on a mobile phone due to obvious reasons. Mobile phones have light-weight implementations of Bitcoin (SPV Wallets) and therefore only ask full node peers for related data.