No it's not possible. Here's why.
First of all let's talk about forks. Like JBaczuk stated, a fork is simply when you copy the code and alter some information. So when people say Dogecoin and Litecoin are forked from Bitcoin, they're saying that they have similar code and that some of the parameters have changed. So even though they are similar, they are different.
Now let's talk about how a blockchain is created. Miners submit a block in accordance to a set of rules. An example of a rule is, like you said, the mining algorithm. Other examples are blocksize limits and blocktimes. For example, Litecoin's blocktime strives for 2.5 minutes while Dogecoin's is 1 minute.
What happens next is this block gets sent out to the network in order to be "validated." This means that full nodes that download the whole blockchain choose to accept or reject this block. Their criteria? The protocol rules of the network. So let's say a miner decides to submit a block that has a 1 minute blocktime (Dogecoin) to a 2.5 minute blocktime network (Litecoin). Even though they share the same mining algo and have similar characteristics, the full nodes in the Litecoin network will reject the 1 minute block and refuse to download it. Instead, they will wait for a block that fits ALL of the criteria (protocol rules).